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THE  PRINCETON  COLLOQUIUM 

u  J 

LECTURES  ON  MATHEMATICS 


DELIVERED  SEPTEMBER  15  TO  17,  1909,  BEFORE 
MEMBERS  OP  THE  AMERICAN  MATHEMATICAL 
SOCIETY  IN  CONNECTION  WITH  THE  SUMMER 
MEETING  HELD  AT  PRINCETON  UNIVERSITY, 
PRINCETON,  N.  J. 


BY 

GILBERT  AMES  BLISS 

AND 

EDWARD  KASNER 


NEW  YORK 
PUBLISHED  BY  THE 

AMERICAN  MATHEMATICAL  SOCIETY 

501  WEST  116TH  STREET 

1913 


PRESS  OF 

THE  NEW  ERA  PRINTING  COMPANY 
LANCASTER.  PA 


PREFACE. 

Soon  after  its  expansion  in  1894  into  a  national  organization, 
the  American  Mathematical  Society  inaugurated  the  series  of 
Colloquia  which  have  been  held  in  connection  with  its  summer 
meetings  since  1896,  at  intervals  of  two  or  three  years.  These 
Colloquia  consist  of  courses  of  lectures  delivered  by  specialists 
on  selected  chapters  of  their  fields  of  work.  Their  purpose  is 
to  enable  the  members  of  the  Society  to  keep  in  touch  with  the 
most  recent  advances  of  mathematical  science  and  to  stimulate 
a  wide  interest  in  its  development. 

The  list  of  Colloquia  thus  far  held  is  as  follows : 

I.  THE  BUFFALO  COLLOQUIUM,     1896. 

(a)     Professor  MAXIME  BOCHER,  of  Harvard  University :  "  Linear 

Differential  Equations,  and  Their  Applications." 
This  Colloquium  has  not  been  published,  but  several  papers 
appeared  at  about  the  time  of  the  Colloquium,  in  which  the 
author  dealt  with  topics  treated  in  the  lectures.* 

(6)     Professor  JAMES  PIERPONT,  of  Yale  University:  "Galois's 

Theory  of  Equations." 

Published  in  the  Annals  of  Mathematics,  series  2,  volumes  1 
and  2  (1900). 

II.  THE  CAMBRIDGE  COLLOQUIUM,    1898. 

(a)   Professor   WILLIAM   F.   OSGOOD,   of   Harvard   University: 

"Selected  Topics  in  the  Theory  of  Functions." 
Published  in  the  Bulletin  of  the  American  Mathematical  Society, 
volume  5  (1898),  pages  59-87. 

*Two  of  these  papers  were:  "Regular  points  of  linear  differential  equa- 
tions of  the  second  order,"  Harvard  University,  1896;  "  Notes  on  some  points 
in  the  theory  of  linear  differential  equations,"  Annals  of  Mathematics,  vol. 
12  (1898). 

i 


11  PREFACE. 

(6)  Professor  ARTHUR  G.  WEBSTER,  of  Clark  University:  "The 
Partial  Differential  Equations  of  Wave  Propagation." 

III.  THE  ITHACA  COLLOQUIUM,     1901. 

(a)     Professor  OSKAR  BOLZA,  of  the  University  of  Chicago :  "  The 
Simplest  Type  of  Problems  in  the  Calculus  of  Variations." 
Published  in  amplified  form  under  the  title:  Lectures  on  the 
Calculus  of  Variations,  Chicago,  1904. 

(6)  Professor  ERNEST  W.  BROWN,  of  Haverford  College:  "Mod- 
ern Methods  of  Treating  Dynamical  Problems,  and  in 
Particular  the  Problem  of  Three  Bodies." 

IV.  THE  BOSTON  COLLOQUIUM,     1903. 

(a)  Professor  HENRY  S.  WHITE,  of  Northwestern  University: 
"Linear  Systems  of  Curves  on  Algebraic  Surfaces." 

(6)  Professor  FREDERICK  S.  WOODS,  of  the  Massachusetts  Insti- 
tute of  Technology:  "Forms  of  Non-Euclidean  Space." 

(c)     Professor  EDWARD  B.  VAN  VLECK,  of  Wesleyan  University: 
"Selected  Topics  in  the  Theory  of  Divergent  Series  and 
Continued  Fractions." 
This  Colloquium  was  published  for  the  Society  in  the  volume: 

The  Boston  Colloquium  Lectures  on  Mathematics,  New  York, 

Macmillan,  1905. 

V.  THE  NEW  HAVEN  COLLOQUIUM,     1906. 

(a)  Professor  ELIAKIM  H.  MOORE,  of  the  University  of  Chicago : 
"On  the  Theory  of  Bilinear  Functional  Operations." 

(6)  Professor  ERNEST  J.  WILCZYNSKI,  of  the  University  of  Cali- 
fornia: "Projective  Differential  Geometry." 

(c)  Professor  MAX  MASON,  of  Yale  University :  "  Selected  Topics 
in  the  Theory  of  Boundary  Value  Problems  of  Differential 
Equations." 


PREFACE.  Ill 

Published  by  Yale  University  in  the  volume:  The  New  Haven 
Mathematical  Colloquium,  New  Haven,  Yale  University  Press, 
1910. 

VI.  THE  PRINCETON  COLLOQUIUM,    1909. 

(a)     Professor  GILBERT  A.  BLISS,  of  the  University  of  Chicago : 
"Fundamental  Existence  Theorems." 

(6)     Professor  EDWARD  KASNER,  of  Columbia  University:  "Dif- 
ferential-Geometric Aspects  of  Dynamics." 
This  Colloquium  is  published  here  in  full. 

The  Colloquia  of  the  Society  are  to  an  extent  comparable  with 
the  reports  regularly  presented  to  Section  A  of  the  British  Associa- 
tion for  the  Advancement  of  Science  and  to  the  Deutsche  Mathe- 
matiker-Vereinigung,  and  in  so  far  play  a  role  complementary  to 
those  of  the  Bulletin  and  Transactions.  The  Society  will  doubt- 
less adopt  the  custom  of  publishing  the  lectures  of  each  Colloquium 
in  a  corresponding  volume. 

The  Seventh  Colloquium  will  be  held  in  connection  with  the 
twentieth  summer  meeting  of  the  Society  at  Madison,  Wisconsin 
during  the  week  September  8-13,  1913.  Courses  of  lectures  will 
be  given  by  Professor  LEONARD  E.  DICKSON,  of  the  University 
of  Chicago,  and  Professor  WILLIAM  F.  OSGOOD,  of  Harvard 
University.  Thus  for  the  first  time  an  interval  of  four  years  has 
elapsed  between  successive  Colloquia.  As  a  suitable  reflection 
and  desirable  stimulation  of  the  mathematical  activity  of  this 
country,  it  would  seem  desirable  that  the  Colloquia  should  be 
held  oftener.  To  avoid  collision  with  the  meetings  of  the  Inter- 
national Congress  of  Mathematicians,  the  Colloquia  might  per- 
haps be  arranged  for  every  odd  numbered  year. 

E.  H.  MOORE. 


FUNDAMENTAL  EXISTENCE 
THEOREMS 


BY 

GILBERT  AMES  BLISS 


CONTENTS 


Pages 

INTRODUCTION 1 

CHAPTER  I 
ORDINARY  POINTS  OF  IMPLICIT  FUNCTIONS 

1.  The  fundamental  theorem 7 

2.  Equations  in  which  the  functions  are  analytic 12 

3.  Goursat's  method  of  approximation 16 

4.  Bolza's  extension  of  the  fundamental  theorem 19 

5.  The  unique  sheet  of  solutions  associated  with  an  initial 

solution 21 

6.  Auxiliary  theorems  and  definitions 28 

7.  A  criterion  that  a  sheet  of  solutions  be  single- valued .  .  33 

8.  Transformations  of  n  variables  and  a  modification  of 

a  theorem  of  Schoenflies 37 

CHAPTER  II 
SINGULAR  POINTS  OF  IMPLICIT  FUNCTIONS 

Introduction 43 

9.  The  preparation  theorem  of  Weierstrass 49 

10.  The  zeros  of  <p(u,  t?),  $(u,  v),  or  their  functional  deter- 

minant       53 

11.  Singular  points  of  a  real  transformation  of  two  variables     60 

12.  The  case  where  the  functional  determinant  vanishes 

identically 67 

13.  A  generalization  of  the  preparation  theorem  of  Weier- 

strass       70 

14.  Applications  of  the  preceding  theory    78 

i 


11  CONTENTS. 

CHAPTER  III 

EXISTENCE  THEOREMS  FOR  DIFFERENTIAL  EQUATIONS 

Introduction 86 

15.  The  convergence  inequality 88 

16.  The  Cauchy   polygons  and   their  convergence  over  a 

limited  interval 89 

17.  The  existence  of  a  solution  extending  to  the  boundary 

of  the  region  R 93 

18.  The  continuity  and   differentiability  of  the  solutions    95 

19.  An  existence  theorem  for  a  partial  differential  equation 

of  the  first  order  which  is  not  necessarily  analytic .  .     98 


BY 

GILBERT  AMES  BLISS 


INTRODUCTION 

The  existence  theorems  to  which  these  lectures  are  devoted 
have  been  the  subject  of  a  long  sequence  of  investigations 
extending  from  the  time  of  Cauchy  to  the  present  day,  and 
have  found  application  at  the  basis  of  a  variety  of  mathematical 
theories  including,  as  perhaps  of  especial  importance,  the  theory 
of  algebraic  functions  and  the  calculus  of  variations.  If  a  single 
solution  (a;  6)  =  (ai,  «2,  •••,  am;  b\,  b%,  •••,  bn)  of  a  set  of 
equations 

/.(*i,  *2,  •  • •,  xn',  yi,  1/2,  •  •  •,  yn)  =  0     (a  =  1,  2,  •  •  •,  n) 

is  known,  then  in  a  neighborhood  of  (a  ;  b)  there  is  one  and  only 
one  other  solution  corresponding  to  each  set  of  values  z  in  a 
properly  chosen  neighborhood  of  the  values  a,  and  in  the  totality 
of  solutions  (x ;  y)  so  defined  the  variables  y  are  single-valued 
and  continuous  functions  of  the  x's.  If  a  set  of  initial  constants 
(£>  ^b  772,  •  •  •,  r]n)  is  given,  then  in  a  neighborhood  of  these  values 
there  is  one  and  but  one  continuous  arc 

ya  =  ya(x)  (a  =  1,2,  •••,») 

satisfying  the  differential  equations 

dya 

-fa  =  9*(x>  Vi>  2/2,  •  •  •,  y»)    (a  =  1,  2,  •  •  •,  n) 

and  passing  through  the  initial  values  77  when  x  =  £. 
2  1 


2  THE   PRINCETON   COLLOQUIUM. 

The  formulation  and  first  satisfactory  proofs  of  these  theorems, 
at  least  for  the  case  where  only  two  variables  x,  y  are  involved, 
seem  to  be  ascribed  with  unanimity  to  Cauchy.  For  the  implicit 
functions  his  proof  rested  upon  the  assumption  that  the  function 
/  should  be  expressible  by  means  of  a  power  series,  and  the 
solution  he  sought  was  also  so  expressible,  a  restriction  which 
was  later  removed  with  remarkable  insight  by  Dini.  For  a 
differential  equation,  on  the  other  hand,  Cauchy  assumed  only 
the  continuity  of  the  function  g  and  its  first  derivative  for  y, 
and  his  method  of  proof,  with  the  well-known  alteration  due  to 
Lipschitz,  retains  to-day  recognized  advantages  over  those  of 
later  writers. 

In  the  following  pages  (§§1,  16)  the  two  theorems  stated 
above  are  proved  with  such  alterations  in  the  usual  methods  as 
seemed  desirable  or  advantageous  in  the  present  connection. 
The  proof  given  for  the  fundamental  theorem  of  implicit  functions 
is  applicable  when  the  independent  variables  x  are  replaced  by  a 
variable  p  which  has  a  range  of  much  more  general  type  than  a 
set  of  points  in  an  m-dimensional  z-space.*  It  is  not  necessary 
always  to  know  an  initial  solution  in  order  that  others  may  be 
found.  In  the  treatment  of  Kepler's  equation,  for  example,  which 
defines  the  eccentric  anomaly  of  a  planet  moving  in  an  elliptical 
orbit  in  terms  of  the  observed  mean  anomaly,  one  starts  with  an 
approximate  solution  only  and  determines  an  exact  solution  by 
means  of  a  convergent  succession  of  approximations.  This 
procedure  is  closely  allied  to  a  method  of  approximation  due  to 
Goursat  (§3),  suggested  apparently  by  Picard's  treatment  of  the 
existence  theorem  for  differential  equations. 

One  of  the  principal  purposes  of  the  paragraphs  which  follow, 
however,  is  to  free  the  existence  theorems  as  far  as  possible  from 

*  The  notion  of  a  general  range  has  been  elucidated  by  Moore,  The  New 
Haven  Mathematical  Colloquium,  page  4,  the  special  cases  which  he  partic- 
ularly considers  being  enumerated  on  page  13.  An  application  of  the  method 
of  §  1  of  these  lectures  when  the  range  of  p  is  a  set  of  continuous  curves,  has 
been  made  by  Fischer,  "A  generalization  of  Volterra's  derivative  of  a  function 
of  a  line,"  Dissertation,  Chicago  (1912). 


FUNDAMENTAL   EXISTENCE   THEOREMS.  3 

the  often  inconvenient  restriction  which  is  implied  by  the  words 
"  in  a  neighborhood  of,"  or  which  is  so  aptly  expressed  in  German 
by  the  phrase  "  im  Kleinen."  It  is  evident  from  very  simple 
examples  that  the  totality  of  solutions  (x;  y)  associated  con- 
tinuously with  a  given  initial  solution  of  a  system  of  equations 
/  =  0  of  the  form  described  above,  can  not  in  general  have  the 
property  that  the  variables  y  are  everywhere  single-valued 
functions  of  the  variables  x,  and  the  result  of  attempting, 
perhaps  unconsciously,  to  preserve  the  single-valued  character 
of  the  solutions  has  been  the  restriction  of  the  region  to  which  the 
existence  theorems  apply.  In  order  to  avoid  this  difficulty  and 
to  characterize  to  some  extent  the  totality  of  solutions  associated 
continuously  with  a  given  initial  one  in  a  region  specified  in 
advance,  the  writer  has  introduced  (§5)  the  notion  of  a  particular 
kind  of  point  set  called  a  sheet  of  points.  In  a  suitably  chosen 
neighborhood  of  a  point  (a;  6)  of  the  sheet  there  corresponds 
to  every  set  of  values  x  sufficiently  near  to  the  values  a  exactly 
one  point  (x;  y)  of  the  sheet,  and  the  single- valued  functions. 
y  so  determined  are  continuous  and  have  continuous  first  de- 
rivatives. This  condition  does  not  at  all  imply  that  there  are 
no  other  points  of  the  sheet  outside  the  specified  neighborhood 
of  the  point  (a;  b)  and  having  a  projection  x  near  to  a.  With 
the  help  of  the  notion  of  a  sheet  of  points  it  can  be  concluded  that 
with  any  initial  solution  (a;  6)  of  the  equations  /  =  0  there  is 
associated  a  unique  sheet  »S  of  solutions  whose  only  boundary 
points  are  so-called  exceptional  points  where  the  functions  / 
either  actually  fail,  or  else  are  not  assumed,  to  have  the  continuity 
and  other  properties  which  are  demanded  in  the  proof  of  the 
well-known  theorem  for  the  existence  of  solutions  in  a  neighbor- 
hood of  an  initial  one.  It  is  important  oftentimes  to  know 
whether  or  not  a  sheet  of  solutions  is  actually  single-valued 
throughout  its  entire  extent,  and  a  criterion  sufficient  to  ensure 
this  property  has  also  been  derived  (§  7). 

On  the  basis  of  these  results  some  important  theorems  con- 
cerning  the   transformation   of   plane   regions   into   regions   of 


4  THE   PRINCETON  COLLOQUIUM. 

another  plane  by  means  of  equations  of  the  form 

*i  =  t\(y\,  2/2),       3-2  =  ^2(2/1, 2/2), 

as  in  the  theory  of  conformal  transformation,  have  been  deduced 
(§8).  If  the  functions  ^  have  suitable  continuity  properties 
and  a  non-vanishing  functional  determinant  in  the  interior  of  a 
simply  closed  regular  curve  B  in  the  z/-plane,  and  if  B  is  trans- 
formed into  a  simply  closed  regular  curve  A  of  the  .r-plane,  then 
the  equations  define  a  one-to-one  correspondence  between  the 
interiors  of  A  and  B,  and  the  inverse  functions  so  defined  have 
continuity  properties  similar  to  those  of  \f/i  and  ^2-  This  is  but 
a  sample  of  the  theorems  which  may  be  stated.  Others  are  also 
given  (§  8)  which  apply  to  the  transformation  of  regions  not 
necessarily  finite,  and  to  systems  containing  more  than  two 
equations. 

The  theory  of  the  singularities  of  implicit  functions  is  of  con- 
siderable difficulty  and  has  been  but  incompletely  developed. 
For  a  transformation  of  the  form  above  in  which  the  functions 
^i,  ^2  are  analytic,  the  singular  point  to  be  studied,  at  which  the 
functional  determinant  D  =  d($\,  4/z)/d(yi,  2/2)  vanishes,  as 
well  as  its  image  in  the  .r-plane,  may  both  without  loss  of  gener- 
ality be  supposed  at  the  origin.  The  most  general  case  under 
these  circumstances  is  that  for  which  the  determinant  D  does 
not  vanish  identically  and  the  equations  \f/i  =  0,  ^2  =  0  have  no 
real  solutions  in  common  near  the  origin  except  the  values 
yl  =  y2  =  0  themselves.  It  is  found  that  the  branches  of  the 
curve  D  =  0  bound  off  with  a  suitably  chosen  circle  about  the 
origin  a  number  of  triangular  regions.  Each  of  these  regions  is 
transformed  in  a  one-to-one  way  into  a  sort  of  Riemann  surface 
on  the  z-plane  which  winds  about  the  origin  and  is  bounded  by 
the  image  of  the  boundary  of  the  triangular  region  (see  §11, 
Fig.  6).  If  the  signs  of  D  in  two  adjacent  triangular  regions 
are  opposite,  then  their  images  overlap  along  the  common 
boundary;  otherwise  they  adjoin  without  overlapping.  At  any 
point  of  one  of  the  Riemann  surfaces  the  inverse  functions  defined 


FUNDAMENTAL   EXISTENCE  THEOREMS.  5 

by  the  transformation  are  continuous  and  in  the  interior  of  the 
surface  they  have  everywhere  continuous  derivatives.  These 
results  are  obtained  by  means  of  applications  of  the  theorem 
described  above  for  the  transformation  of  the  interior  of  a  simply 
closed  curve  B;  and  the  same  method  of  procedure  would  un- 
doubtedly be  of  service  when  the  curves  \f/i  =  0,  ^2  =  0  have 
real  branches  through  the  origin  in  common,  which  must  occur 
whenever  they  have  common1  points  in  every  neighborhood  of 
the  values  y\  =  yi  =  0.  The  case  where  the  determinant  D 
vanishes  identically  is  also  considered  (§  12). 

For  the  singularities  of  implicit  functions  defined  by  a  sys- 
tem of  equations  /  =  0  there  is  a  generalization  of  the  prepara- 
tion theorem  of  Weierstrass  (§  9)  suggested  to  the  writer  by 
some  remarks  in  the  introduction  of  Poincare's  Thesis,  and 
by  a  study  of  the  elimination  theory  of  Kronecker  for  algebraic 
equations.  The  theorem  is  presented  here  (§13)  for  two  equa- 
tions and  two  variables  y\,  y-i  in  the  form  originally  given  at  the 
time  of  the  Princeton  Colloquium,  but  the  method  of  proof  is 
similar  to  that  of  a  later  paper*  and  applies  with  suitable  modi- 
fications to  a  system  containing  more  equations  and  independent 
variables.  These  results  can  not  by  any  means  be  said  to  afford 
a  complete  characterization  of  the  singularities  of  implicit 
functions,  but  it  is  hoped  that  they  may  be  useful  in  paving  the 
way  for  researches  of  a  more  comprehensive  character. 

The  writer  published  some  years  ago  a  paper  f  concerning  the 
extensibility  of  the  solutions  of  a  system  of  differential  equations, 
of  the  form  specified  above,  from  boundary  to  boundary  of  a  finite 
closed  region  R  in  which  the  functions  ga  are  supposed  to  have  suit- 
able continuity  properties.  In  the  last  chapter  of  these  lectures  the 
character  of  the  region  has  been  generalized  so  that  no  restrictions 
as  to  its  finiteness  or  closure  are  made,  and  it  is  shown  that  the 
approximations  of  Cauchy  converge  to  a  solution  over  an  interval 


*  See  the  footnote  to  page  73. 

t  "  The  solutions  of  differential  equations  of  the  first  order  as  functions  of 
their  initial  values,"  Annals  of  Mathematics,  2d  series,  vol.  6  (1904),  page  49. 


6  THE   PRINCETON   COLLOQUIUM. 

in  the  interior  of  which  the  limiting  curve  is  continuous  and 
interior  to  R,  while  at  the  ends  of  the  interval  the  only  limit 
points  of  the  curve  are  at  infinity  or  else  are  on  the  boundary  of  the 
region.  The  solutions  so  defined  are  continuous  and  differenti- 
able  with  respect  to  their  initial  values,  a  property  which  once 
proved  is  of  great  service  in  many  of  the  applications  of  the 
existence  theorems.  One  situation  in  which  these  results  have 
an  important  bearing  is  related  to%  partial  differential  equation 
of  the  first  order 

F(x,  y,  z,  dz/dx,  dz/dy)  =  0. 

When  this  equation  is  analytic,  any  analytic  curve  C,  which  is 
not  a  so-called  integral  curve,  defines  uniquely  an  analytic  surface 
containing  the  curve  and  satisfying  the  differential  equation.  The 
uniqueness  in  this  case  is  a  consequence,  in  the  first  place,  of 
the  fact  that  an  analytic  surface  is  completely  determined  when 
an  initial  series  defining  its  values  in  a  limited  region  is  given, 
and,  in  the  second  place,  of  the  theorem  that  at  a  given  point 
and  normal  of  the  initial  curve  C  satisfying  the  differential  equa- 
tion there  is  but  one  series  defining  an  integral  surface  including 
the  points  of  C  and  having  the  given  initial  normal.  It  is  not 
self  evident  in  what  sense  a  solution  of  a  non-analytic  equation 
is  uniquely  determined  by  an  initial  curve,  as  may  be  seen  by  very 
simple  examples.  An  initial  curve  which  is  not  an  integral  curve 
will  in  general  have  associated  with  it,  however,  a  strip  of  nor- 
mals which  satisfy  the  partial  differential  equation,  and  whose 
elements  as  initial  values  determine  a  one-parameter  family  of 
characteristic  strips  simply  covering  a  region  Rxy  of  the  xy-p\ane 
about  the  projection  of  the  initial  curve  C.  There  is  one  and  but 
one  integral  surface  of  the  differential  equation  with  a  continu- 
ously turning  tangent  plane  and  continuous  curvature,  which  is 
defined  at  every  point  of  the  region  Rxy  and  contains  the  initial 
curve  C  and  its  strip  of  normals  (§  19). 


CHAPTER  I 

ORDINARY    POINTS    OF    IMPLICIT    FUNCTIONS 

§  1.    THE  FUNDAMENTAL  THEOREM 

The  fundamental   theorem    of  the   implicit  function   theory 
states  the  existence  of  a  set  of  functions 


which  satisfy  a  system  of  equations  of  the  form 

(1)        fa(xi,  :r2,  •  •  -,  .rm;  j/i,  z/2,  •  •  •  ,  2/n)  =  0      (a  =  1,  2,  •  •  •,  ri) 

in  a  neighborhood  of  a  given  initial  solution  (a;  6).  Dini's 
method,*  for  the  case  in  which  the  functions  /are  only  assumed  to 
be  continuous  and  to  have  continuous  first  derivatives,  is  to 
show  the  existence  of  a  solution  of  a  single  equation,  and  then 
to  extend  his  result  by  mathematical  induction  to  a  system  of 
the  form  given  above,  a  plan  which  has  been  followed,  with 
only  slight  alterations  and  improvements  in  form,  by  most 
writers  on  the  theory  of  functions  of  a  real  variable.  In  a  more 
recent  paperf  Goursat  has  applied  a  method  of  successive  ap- 
proximations which  enabled  him  to  do  away  with  the  assumption 
of  the  existence  of  the  derivatives  of  the  functions  /  with  respect 
to  the  independent  variables  x. 

One  can  hardly  be  dissatisfied  with  either  of  these  methods  of 
attack.  It  is  true  that  when  the  theorem  is  stated  as  precisely 
as  in  the  following  paragraphs,  the  determination  of  the  neighbor- 
hoods at  the  stage  when  the  induction  must  be  made  is  rather 
inelegant,  but  the  difficulties  encountered  are  not  serious.  The 
introduction  of  successive  approximations  is  an  interesting  step, 

*  Lezioni  di  Analisi  infinitesimale,  vol.  1,  chap.  13.  For  historical  remarks, 
see  Osgood,  Encyclopadie  der  mathematischen  Wissenschaften,  II,  B  1,  §  44 
and  footnote  30. 

If  Bulletin  de  la  Societe  mathematique  de  France,  vol.  31  (1903),  page  185. 

7 


8  THE   PRINCETON   COLLOQUIUM. 

though  it  does  not  simplify  the  situation  and  indeed  does  not 
add  generality  with  regard  to  the  assumptions  on  the  functions  /. 
The  method  of  Dini  can  in  fact,  by  only  a  slight  modification, 
be  made  to  apply  to  cases  where  the  functions  do  not  have 
derivatives  with  respect  to  the  variables  x.  The  proof  which  is 
given  in  the  following  paragraphs  seems  to  have  advantages  in 
the  matter  of  simplicity  over  either  of  the  others.  It  applies 
equally  well,  without  induction,  to  one  or  a  system  of  equations, 
and  requires  only  the  initial  assumptions  which  Goursat  mentions 
in  his  paper. 

Where  it  is  possible  without  sacrificing  clearness,  the  row  letters 
/,  x,  y,  p,  a,  b'  will  be  used  to  denote  the  systems 

/  =    (/l,/2,    '  '  -,/n),  X  =    (Xi,  *2,    '  '  ',  Xm), 

y  =  (y\,  2/2,  •  •  •,  yn),        a  =  (ai,  a2,  •  •  •,  dm), 

b  =  (bi,  62,  •  •  •,  bn),        p  =  (ai,  a2,  •  •  •,  am;  bi,  bz,  •••,  &„). 

In  this  notation  the  equations  (1)  have  the  form 

/(*;  y)  =  o, 

the  interpretation  being  that  every  element  of  /  is  a  function  of 
xi,  x2,  —  -,  xm;  j/i,  j/2,  •  •  •,  yn,  and  every  /,-  is  to  be  set  equal  to 
zero.  The  notations  pt,  at}  bt  represent  respectively  the  neigh- 
borhoods 


x  —  a 


<  e,       y  —  b    <*;       x  —  a     <  e;     \y  —  b\  <  e 


of  the  points  p,  a,  b. 

With  these  notations  in  mind  the  fundamental  theorem  which 
is  to  be  proved  may  be  stated  as  follows: 

Hypotheses  : 

1)  the  functions  f(x;  y)  are  continuous,  and  have  first  partial 
derivatives  with  respect  to  the  variables  y  which  are  also  continuous, 
in  a  neighborhood  of  the  point  (a;  6)  ichich  will  be  denoted  by  p; 

2)  /(a;  6)  =  0; 

3)  the  functional  determinant  D  =  d(fi,  /2,    •  •  •,  fn)/d(yi,  yz, 
•  •  •  ,  J/n)  is  different  from  zero  at  p. 


FUNDAMENTAL   EXISTENCE   THEOREMS.  9 

Conclusions  : 

1)  a  neighborhood  pt  can  be  found  in  which  there  corresponds 
to  a  given  value  x  at  most  one  solution  (x;  y}  of  the  equations 

/(*;  y)  =  0; 

2)  for  any  neighborhood  pt  with  the  property  just  described  a 
constant  5  ^  e  can  be  found  such  that  every  x  in  as  has  associated 
with  it  a  point  (x;  y)  which  satisfies  the  equations  f(x;  y)  =  0; 

3)  the  functions  y(x\,  #2,  •  •  «,  xm}  so  found  are  continuous  in 
the  region  as. 

For  the  neighborhood  pf  let  one  be  chosen  in  which  the 
continuity  properties  of  the  functions  /  are  preserved.  If 
(x;  y)  and  (x;  y')  are  two  points  in  pt,  it  follows,  by  applying 
Taylor's  formula  to  the  differences  f(x  ;  y'}  —  f(x;  y),  that 

A(*;  y')  -  /i(*;  y)  =      (y/  -*)+•••+      (y-'  -  y»), 


/n(s;  y')  -/n(z;y)  =  ^  (2/1'  -  yO  +  ----  h  ^7  (yn'  -  yB), 

where  the  arguments  of  the  derivatives  dfjdy$  have  the  form 
z;  V  +  Qa.(y'  ~  y},  and  0  <  0a  <  1.  The  determinant  of  these 
derivatives  is  different  from  zero  when  (x;  y'}  =  (x;  y}  =  (a;b), 
and  hence  must  remain  different  from  zero  if  p€  is  restricted  so 
that  in  it  the  functional  determinant  D  remains  different  from 
zero.  It  is  then  impossible  that  (x;  y}  and  (x;  y')  should  both 
be  solutions  of  the  equations  f(x;  y)  =  0,  if  y  is  distinct  from  y'. 
In  the  corresponding  region  bf  the  function 

<p(&;  y)  =  /i20;  y)  +/22(a;  y)  +  •  •  •  +/n2(a;  y) 

has  a  minimum  for  y  =  b,  since  for  that  value  it  vanishes  and 
for  every  other  it  is  positive.     In  particular 

(p(a\  77)  —  (p(a;  6)  >  m  >  0 

when  77  ranges  over  the  closed  set  of  points  77  forming  the  boundary 
of  bf,  on  account  of  the  continuity  of  <p,  and  the  inequality 

<p(x;  17)  —  <p(x;  6)  >  m 


10  THE   PRINCETON  COLLOQUIUM. 

remains  true  for  all  values  x  in  a  suitably  chosen  domain  a&. 
Hence  for  a  fixed  x  in  as  the  minimum  of  <p(x;  y)  is  attained  at  a 
point  y  interior  to  bt.  At  such  a  point,  however, 

Id?  'dh, 


i  a* 


and  this  can  happen  only  when  all  the  elements  of  /  are  zero, 
since  the  functional  determinant  D  is  different  from  zero  in  pt. 
It  follows  that  to  every  point  x  in  a&  there  corresponds  in  p( 
a  solution  (x;  y)  of  the  equations  /(a*;  y)  =  0. 

The  functions  y(x\,  x%,  •  •  -,  xm}  defined  in  this  way  over  the 
region  a&  are  all  continuous.     For  consider  the  values  y  and 
y  +  Ay  corresponding  to  two  points  x  and  x  +  A.r.     By  apply- 
ng  Taylor's  formula  it  follows  from  the  relations 

f(x;  y  +  Ay)  -  f(x;  y}  =  f(x;  y  +  Ay)  -  f(x  +  A*;  y  +  Ay), 

which  are  true  because  (#;  y)  and  (a:  +  Ax;  y  -f-  Ay)  both  make 
/  =  0,  that 


=  /i(z;  y  +  Ay)  -  /!(*  -f  Aar;  y  +  Ay), 
(2)       .......... 


=  fn(x;  y  H-  Ay)  -  /n(.r  +  Ax;  y  +  Ay), 

where  the  arguments  of  the  derivatives  dfjdyft  have  the  form 
*;  y  H-  0aAy  (0  <  0a  <  1).  The  determinant  of  these  deriv- 
atives is  different  from  zero  on  account  of  the  way  in  which  pt 
was  chosen,  and  the  second  members  of  the  equations  approach 
zero  with  A.r.  Hence  the  same  must  be  true  of  the  quantities 


FUNDAMENTAL  EXISTENCE  THEOREMS.  11 

Ay,  and  thus  the  functions  y(xi,  ar2,  •  •  •  ,  xm)  are  seen  to  be 
continuous. 

A  similar  application  of  Taylor's  formula  leads  to  the  con- 
clusion: 

//  the  functions  f  have  derivatives  of  the  first  order  with  respect 
to  Xk  which  are  continuous  in  the  neighborhood  of  p,  so  have  also 
the  functions  y(x\,  .T2,  •••,  xm}  in  the  region  as',  and  if  the  f's 
have  all  derivatives  of  the  nth  order  continuous,  so  have  the  functions 
y(xi,  x2,  •••,  xm}. 

For  suppose 

Azi  4=  0,    Az2  =  Aa*3  =  •  •  •  =  AzTO  =  0. 

Then  by  applying  Taylor's  formula  to  the  second  members  of 
equations  (2)  it  follows  that 

Ayn 


/i  ,i/2  i     yn  i=  Q 


,     n*  i   ....     ^        .     »  =  0 


where  the  arguments  of  the  derivatives  dfjdxi  have  the  form 
x  +  6a'A.x;  y  -\-  A?/.  Hence  as  A#i  approaches  zero  the  quotients 
approach  limits  dyjdxi  which  satisfy  the  equations 


<±<yj,,i<y?+  ...  ,     L  y»  ,       =  0 

dyidxi       dyzdxi  dyndxi      dxi 

(3)  •*       • 

dfndyi.dfndyz  dfn  dyn      dfn  = 

dyidxi       dyzdxi  dyndxi      dxi~ 

where  the  arguments  of  the  derivatives  of  /  are  now  (x;  y). 
A  similar  consideration  shows  the  existence  of  the  first  deriv- 
atives with  respect  to  the  variables  x2,  xz,  •  •  •  ,  xm.  The  ex- 
istence of  the  higher  derivatives  follows  from  the  observation 
that  the  solutions  of  equations  (3)  for  the  quotients  dfjdy$  are 


12  THE   PRINCETON   COLLOQUIUM. 

differentiable  n  —  1  times  with  respect  to  the  variables  x,  on 
account  of  the  assumption  that  the  functions  /  are  differentiable 
n  times. 

§  2.    EQUATIONS  IN  WHICH  THE  FUNCTIONS  ARE  ANALYTIC 

It  seems  necessary  to  proceed  differently  in  order  to  prove  that 
when  the  functions /in  equations  (1)  are  analytic  with  coefficients 
and  variables  permitted  to  assume  imaginary  values,  the  solutions 
y  =  y(xi,  Xz,  • '  -,  xm)  are  also  analytic  functions  of  the  variables 
x.  The  following  theorem  can  first  be  proved : 

When  the  functions  f  are  formal  series  in  the  variables  x;  y  with 
literal  coefficients  and  having  no  constant  terms,  then  there  exists 
one  and  but  one  set  of  series 

(4)  ya  =  ya(xlyx^  •••,a-TO) 

for  the  variables  y,  which  vanish  with  the  x  s  and  satisfy  identically 
the  equations  f(x;  y)  =  0.  Each  coefficient  in  the  series  y  is 
rational  in  a  finite  number  of  those  of  the  functions  f,  the  only 
denominators  occurring  being  powers  of  the  determinant  R  of  the 
coefficients  of  the  linear  terms  in  y. 

To  prove  this  let  the  equations  /  =  0  be  written  in  the  form 

a\\yi  +  any2  +  •  •  •  +  ainyn  =  g\(x;  y}, 

an\yi  +  «n2i/2  -h  •  •  •  +  annyn  =  gn(x;  y), 

where  the  functions  g  have  no  linear  terms  in  y.  By  multiplying 
these  equations  by  proper  factors  and  adding,  they  may  be  made 
to  take  the  form 

(5)  ya  =  ha(x;  y)  (a  =  1,  2,  •  •  •,  n), 

where  the  series  h  have  still  no  linear  terms  in  y  and  have  coeffi- 
cients which  are  rational  in  those  of  the  functions  /,  the  only 
denominators  occurring  being  the  determinant  R.  Any  series 
for  y  which  satisfy  formally  the  original  equations  must  satisfy 
the  last  equations,  and  vice  versa. 


FUNDAMENTAL  EXISTENCE  THEOREMS.  13 

Consider  now  a  set  of  series  (4)  in  which  the  coefficients  are 
indeterminates  c.  If  they  satisfy  the  equations  (5)  identically, 
then  by  comparison  of  coefficients  on  the  two  sides  it  is  seen 
that  any  coefficient  cv  of  a  term  of  degree  v  must  be  equal  to  a 
polynomial,  with  positive  integral  coefficients,  in  a  finite  number 
of  the  coefficients  of  the  functions  h  and  in  the  coefficients  <?„_*; 
of  terms  in  the  functions  y  of  lower  degree  than  v.  For  there 
are  at  most  a  finite  number  of  terms  on  the  right  of  any  given 
degree  v,  and  since  the  functions  h  have  no  linear  terms  in  the 
variables  y  it  follows  that  wherever  the  term  containing  cv 
occurs  it  is  always  multiplied  by  a  y  or  by  a  power  of  some  of 
the  variables  x,  and  hence  cv  can  only  appear  in  terms  of  degree 
greater  than  v.  Since  the  coefficients  of  the  linear  terms  in  the 
functions  y  are  equal  respectively  to  corresponding  coefficients 
in  the  functions  h,  it  follows  by  an  easy  induction  that  every 
coefficient  in  the  functions  y  must  be  a  polynomial  with  positive 
integral  coefficients  in  a  finite  number  of  the  coefficients 
of  the  functions  h.  There  is  evidently  but  one  set  of  series  (4) 
of  the  kind  described  satisfying  formally  the  equations  (5),  or 
what  is  the  same  thing,  the  equations  /  =  0. 

For  any  numerical  choice  of  the  coefficients  of  the  functions  f  in 
the  domain  of  real  or  imaginary  numbers  for  which  the  series  f 
converge  and  the  determinant  R  =  aa$  is  different  from  zero, 
the  series  (4)  for  y  will  also  be  well-determined  and  convergent. 

For,  a  set  of  equations 

(6)  ya=Ha(x;y)  (a  =  1,  2,  -  -  -,  n) 

can  be  constructed  whose  coefficients  are  all  positive  and  greater 
numerically  than  the  corresponding  coefficients  in  the  functions 
h,  and  for  which  the  corresponding  series  y  =  Y(XI,  z2>  •  •  • ,  xm} 
converge.  The  coefficients  in  the  functions  Y  will  be  greater 
numerically  than  the  corresponding  coefficients  of  the  series 
y(xi,  x2,  •  ",  Xm),  and  hence  the  series  y  will  also  converge. 

To  show  this  suppose  that  p  is  a  positive  constant  smaller 
than  the  radii  of  convergence  of  the  functions  h(x;  y).  Then 


14  THE  PRINCETON   COLLOQUIUM. 

the  series  h(p;  p)  are  convergent,  and  each  term  is  numerically 
smaller  than  a  constant  M  chosen  greater  than  the  sum  of  the 
absolute  values  of  the  terms  in  any  one  of  the  series  h(p;  p). 
The  coefficient  of  any  term  in  h(x;  y)  is  less  than  M/p"  where  v 
is  the  degree  of  the  term.  The  series 

= 


are  similar  to  the  series  h(x;  y]  in  the  matter  of  missing  terms, 
and  dominate  them  in  the  manner  described  above,  since  the 
coefficient  of  any  term  of  degree  v  is  M/p"  or  greater. 

The  unique  series  satisfying  equations  (6)  will  evidently  be 
convergent  if  a  convergent  series  u  in  x  can  be  determined 
satisfying 


_ 


_ 


Xm  \  (  1    _   nU\ 


for  then  every  series  y  can  be  put  equal  to  that  series  u.     The 
latter  equation  is  however  a  quadratic  in  u  and  has  the  solution 


4Jfn(p+3fn) 
=  1' 


I 

^ 


vanishing  with  x.     This  will  certainly  be  representable  by  a 
convergent  series  in  x  provided  that 


since  then  the  second  term  under  the   radical   is   numerically 
less  than  unity. 


FUNDAMENTAL  EXISTENCE  THEOREMS.  15 

The  two  theorems  which  have  just  been  proved  enable  one  to 
make  the  following  statement  concerning  the  solutions  whose 
existence  was  proved  in  §  1  : 

//  the  functions  f(x;  y)  are  analytic  in  the  region  pf,  then  the 
solutions  (4)  of  the  equations  f(x;  y]  =  0  are  analytic  at  every  point 
of  the  region  as. 

It  is  only  necessary  to  transform  the  origin  of  coordinates  to 
the  particular  point  (x;  y)  of  the  solution  which  it  is  desired  to 
investigate. 

Furthermore  when  the  domain  in  which  the  equations  /  =  0 
are  to  be  studied  is  the  domain  of  complex  numbers,  a  theorem 
analogous  to  that  of  §  1  may  be  stated. 

//  in  the  domain  of  complex  numbers  the  functions  f(x;  y}  are 
analytic  at  a  point  p(a;  6)  at  which 

f,        i^  n          r»/        t\  I     "(/l>/2>    '''ijn) 

f(a;  6)  =  0,     D(a;  b)  =  —-^    x=n  *  0, 

LtffJ/l,  2/2,    '  '  ',  yn)  J  y=b 

then  there  exists  a  neighborhood  pe  in  which  any  x  corresponds  to  at 
most  one  solution  (x;y),  either  real  or  complex,  of  the  equations 
f(x;  y)  =  0.  For  any  such  choice  of  pf  a  neighborhood  as  (5  ^  e) 
can  be  found  such  that  every  point  x  in  as  has  associated  with  it  a 
solution  (x',  y)  of  the  equations  f  =  0  in  pf,  and  the  values  y  for 
these  solutions  are  defined  by  a  set  of  functions 

(7)  ya  =  ya(xi,  xt,  •  •  -,  xm)       (a  =  1,  2,  •  •  •,  ri) 

which  are  expressible  as  series  in  the  differences  x  —  a  convergent 
in  the  region  a&. 

The  existence  of  the  neighborhood  pf  is  provable  by  the  ar- 
gument used  in  §  1,  since  for  any  two  points  (x;  y}  and  (x;  y') 
in  the  common  domain  of  convergence  of  the  functions  /,  equa- 
tions of  the  form 


(a=  1,2,  •••,«) 
hold,  where  the  coefficient  A^  is  a  convergent  series  in  the  dif- 


16  THE   PRINCETON  COLLOQUIUM. 

ferences  x  —  a,  y  —  b,  yf  —  b  with  constant  term  equal  to  aa/3. 
The  existence  of  the  coefficients  A  can  be  established  by  con- 
sidering two  analogous  terms  in  f(x;  y)  and  f(x;  y').  The 
difference  of  such  a  pair  of  terms  will  always  be  linearly  expressible 
in  terms  of  the  differences 

(y.'  -  to  -  (y.  -  to  =  y.'  -  y.   (o  -  1, 2,  •  •-,  n). 

Furthermore  for  (a-,  ?/,  y')  =  (a,  6,  6)  the  derivative  of  the  first 
member  with  respect  to  yft'  reduces  to  aa/3 ,  while  that  of  the  second 
is  the  constant  term  in  AaB.  Hence  for  these  values  of  the 
variables  the  determinant  Aaft  reduces  to  D(a,  6)  4=  0. 

By  transforming  the  origin  of  coordinates  to  the  point  (a,  6) 
and  applying  the  first  two  theorems  of  this  section,  it  follows  that 
there  exists  a  set  of  convergent  series  (7)  satisfying  the  equations 
/  =  0  identically;  and  for  a  sufficiently  small  region  aa  the 
points  (x;  y}  which  they  define  will  all  lie  in  the  neighborhood  pf. 

§  3.     GOURSAT'S  METHOD  OF  APPROXIMATION 

The  method  of  approximation  which  is  to  be  presented  in  the 
following  paragraphs  is  of  interest  primarily  because  it  affords 
a  direct  method  of  finding  the  values  of  implicit  functions,  and 
justifies  computations  sometimes  used  in  the  applications  of 
the  theory.  In  order  to  exhibit  this  method  suppose  again  that 
the  functions  /  have  the  properties  described  in  the  principal 
theorem  of  §  1,  and  consider  the  following  set  of  equations 
suggested  by  Taylor's  formula: 

fi(x  ;  y)  +  an(2/i'  —  yi)  +  012(2/2'  -  2/2)  + 

+  aln(yn'  —  yn)  =  0, 
(8) 

fn(x  ;  y}  +  ani(yi  —  y\)  +  anz(yz  —  2/2)  + 

+  ann(yn'  —  yn)  =  0, 

in  which  the  coefficient  aa&  is  the  value  of  dfjdy^  at  the  point  p. 
When  solved  for  the  variables  y',  these  equations  take  the  form 

(9)  ya'  =  <Pa(x  \y)  (a  =  1,  2,  •  •  -,  n), 


FUNDAMENTAL   EXISTENCE   THEOREMS.  17 

and  one  verifies  readily  by  substitution  of  these  expressions 
in  equations  (8)  that  the  functions  <p  and  all  of  their  first 
derivatives  with  respect  to  the  elements  of  y  are  continuous  near 
p;  and  at  the  point  p  itself  <pa  has  the  value  ba,  while  all  of  its 
derivatives  with  respect  to  the  y's  vanish. 

A  sequence  of  systems  y(k)  =  (y\(k\  y^k\  •  •  •  ,  yn(k))  beginning 
with  the  set 

y'  -  [<pi(x;b),  <pz(x;b),   •••,  <pn(x;b)} 

can  now  be  defined  by  means  of  the  recursion  formulas  (9),  which 
are  equivalent  to 

(a  =  l,2t.-.,n). 


Letpe  be  any  neighborhood  of  p  in  which  the  continuity  properties 
of  /  are  retained,  and  in  which  the  derivatives  of  <p  remain  nu- 
merically less  than  6/n  where  0  <  6  <  1.  If  the  values  of  x  are 
restricted  to  a  region  as(d  ^  e)  so  small  that  every  element  of 
the  set  y'  satisfies  the  inequality 

(10)  ya'  -ba    <  e(l  -  6), 


then  the  points  (x;  yw)  will  all  lie  in  the  neighborhhood  pf 
and  will  approach  uniformly  a  limiting  point  (x;  y}  which  is  a 
solution  of  the  equations  (1). 

To  prove  these  statements  one  needs  only  to  apply  successively 
the  inequality 


which  follows  readily  by  an  application  of  Taylor's  formula. 
Since  the  inequalities  (10)  hold,  the  last  formula  successively 
applied  shows  that 


Consequently  the  sum  yaw  of  the  first  k  +  1  terms  of  the  series 
(11) 


18  THE   PRINCETON   COLLOQUIUM. 

differs  in  absolute  value  from  ba  by  a  quantity  which  is  less  than 
6(1  -  6) (I  +  6  +  02  + h  0*-')  =  e(l  -  0*)  <  «. 

Hence  the  points  (x\  y)  all  lie  in  the  neighborhood  pf,  and  the 
series  (11)  is  uniformly  convergent  in  the  neighborhood  a&. 

The  limiting  point  (x;  y)  evidently  satisfies  the  equations 
/  =  0.  For  at  every  stage  the  values  (x,  y,  y')  =  (x,  y(k~1),  y(K)) 
satisfy  the  equations  (8),  and  the  first  members  of  these  equations 
approach  uniformly  the  values /(x;  y). 

The  process  of  determining  the  solutions  described  above  is 
evidently  one  of  trial  and  error.  The  values  y  =  b  being  first 
substituted,  the  equations  (9)  determine  approximately  the 
correction  y'  —  b  which  must  be  added  to  6  in  order  to  obtain  a 
solution  for  any  value  of  x  near  to  a.  For  the  values  so  corrected 
the  equations  (9)  give  again  a  new  correction  y"  —  y',  and  so  on. 

It  is  ordinarily  presupposed  that  an  initial  solution  (a;  6)  is  given, 
but  the  process  may  also  lead  to  the  discovery  of  a  solution  in  case  only 
an  initial  point  which  approximately  satisfies  the  equation  is  known. 
To  show  this  suppose  that  the  functions/  are  continuous  and  have 
continuous  first  partial  derivatives  with  respect  to  the  variables 
y  in  a  closed  region  R  of  points  (x;  y)  in  which  the  functional 
determinant  D(x;y)  is  different  from  zero.  The  functions  (p  in 
equations  (9)  are  to  be  thought  of  as  depending  upon  (x;y), 
and  also  upon  the  variables  (a;  6)  which  enter  in  the  derivatives 
aa/s.  Then  the  expressions  (p(x,  y,  a,  b),  <py(x,  y,  a,  6)  are  con- 
tinuous when  (x;  y),  (a;  6)  lie  in  R,  and  all  of  the  derivatives 
tpy  vanish  identically  when  (x;  y)  =  (a;  &).  The  value  of 
<p(a,  b,  a,  b)  is  not  necessarily  b,  however,  when  (a;  6)  is  not  a 
solution.  Two  positive  constants,  0  <  1  and  e,  can  be  deter- 
mined so  that 

!  <Pv(x,  y,  a,  6)  |  <  Bin 

whenever  (a;  6)  and  (x;  y)  satisfy  the  inequalities 


x  —  a 


e, 


If  now  there  exists  a  point  p(a;b)  for  which  the  neighborhood  pt 


FUNDAMENTAL  EXISTENCE  THEOREMS.  19 

is  entirely  within  R,  and  such  that 

|  <f>(a,  b,  a,  b)  —  b    <  e(l  —  6), 

then  the  sequence  y(k)  defined  converges  uniformly  as  before 
in  a  neighborhood  as  of  the  point  a  and  determines  a  solution 

As  an  example  consider  the  equation 
(12)  y  -  e  sin  y  =  x  (0  <  e  <  1), 

which  in  the  theory  of  elliptic  orbits  determines  the  value  of  the 
eccentric  anomaly  y  in  terms  of  the  mean  anomaly  x.  The  func- 
tion <p  is  in  this  case 

e(sin  y  —  y  cos  6)  +  x 

<p(x,  y,  a,  b)  =  -  r 

1  —  e  cos  b 

and  <f>y  remains  less  than  6  when 

1  -  e  _ 
e 

For  any  given  x  =  a,  a  value  y  =  b  can  be  determined,  by  graph- 
ical methods  for  example,  so  that 

i     ,       T  x       ,  |        b  -  e  sin  b  -  a 
<f>(a,  b,  a,  b)  —  b  \  = 


1  —  e  cos  6 

The  process  described  above  therefore  converges  in  a  suitably 
chosen  neighborhood  of  x  =  a,  and  a  solution  of  equation  (12) 
can  be  found  when  an  approximate  solution  only  has  been  de- 
termined in  advance. 

§  4.    BOLZA'S  EXTENSION  OF  THE  FUNDAMENTAL 
THEOREM* 

The  neighborhood  P,  of  a  set  of  points  P  in  the  space  (x;  y) 
is  the  totality  of  points  (x;  y}  which  satisfy  inequalities  of  the 
form 

x  —  a  |  <  e,         \y  —  b\  <  e, 


*  Vorlesungen  iiber  Variationsrechnung,  page  160:  also  Mathematische 
Annalen,  vol.  63  (1906),  page  247.  The  theorem  was  proved  independently 
by  Mason  and  Bliss,  "  Fields  of  extremals  in  space,"  Transactions  of  the 
American  Mathematical  Society,  vol.  11  (1910),  page  326. 


20  THE    PRINCETON   COLLOQUIUM. 

where  (a;  6)  is  some  point  of  P.  The  sets  of  points  (a)  and  (b) 
which  belong  to  points  (a;  6)  of  P  are  the  projections  of  P  in 
the  .r-  and  ?/-spaces,  and  will  be  denoted  by  A  and  B,  respectively. 

The  fundamental  theorem  of  §  1  remains  true  if  in  its  statement 
the  single  point  p  is  replaced  by  a  set  of  points  P  which  is  finite 
and  closed,  and  ichich  furthermore  has  the  property  that  no  two 
distinct  points  (a;  b),  (a'\  b'}  of  P  haw  the  same  projection  a'  =  a. 
According  to  the  conclusions  of  the  theorem  there  exists  then  a 
neighborhood  Pe  in  which  no  two  solutions  of  the  equations  f(x;  ?/)  =  0 
have  the  same  projection  x,  and  a  neighborhood  As  in  which  every  x 
surely  belongs  to  a  solution  (x;  y)  in  Pe.  The  single-valued  functions 
y(xi,  a-2,  •  •  •,  Xm)  so  defined  in  As  are  continuous,  and  if  the  func- 
tions f(x;  y]  have  continuous  derivatives  of  the  n-th  order  in  a 
neighborhood  of  P,  so  have  the  functions  y(xi,  .r2,  •••,  xm}  in  A$. 

To  prove  the  theorem  suppose  first  that  a  sequence  of  positive 
constants  «&  (k  =  I,  2,  •  •  •)  approaching  zero  has  been  selected 
arbitrarily.  If  the  first  part  of  the  theorem  were  not  true,  then 
in  any  neighborhood  Pfk  there  would  be  two  distinct  solutions 
(.r;  y}k  and  (x;  y'}k  of  the  equations  f(x;  y)  =  0,  which  would 
satisfy,  respectively,  inequalities  of  the  form 

x  —  a    <  €k,       y  —  ft    <  «A-; 

(13) 

x-a'\<fk,    \y'-P'\<  6, 

with  two  points  (a;  0)k  and  (a';  /3')/t  of  the  set  P.  Since  P  is 
finite  and  closed,  the  sequence  of  values  (a,  /3;  a',  /3')*  has  a 
point  of  condensation  (a,  b;  a',  &')  for  which  (a;  6)  and  (a';  b') 
are  both  in  P.  From  the  inequalities  (13)  it  follows  that 
(a,  6;  a',  6')  is  also  a  point  of  condensation  for  the  sequence 
(x,  y;  x,  y')k,  and  therefore  a  and  a'  must  be  the  same.  The 
values  6  and  b'  must  also  be  identical  since  P  contains  only  one 
point  p(a;  6)  with  the  projection  a.  According  to  the  original 
statement  of  the  fundamental  theorem  in  §  1,  a  neighborhood  p( 
can  be  chosen  in  which  no  two  solutions  of  the  equations 
(x;  y)  =  0  have  the  same  projection  x.  Hence  the  existence 
of  the  sequences  (x;  y)k  and  (x;  y')k  with  the  common  point  of 


FUNDAMENTAL   EXISTENCE   THEOREMS.  21 

condensation  (a;  6)  is  contradicted,  and  it  must  always  be 
possible  to  select  a  neighborhood  Pt  in  which  distinct  solutions 
of  the  equations  /  =  0  always  have  distinct  projections  x. 

A  similar  argument  shows  that  a  neighborhood  A&  can  be 
selected  so  that  to  any  point  of  it  there  corresponds  a  solution 
of  the  equations  /  =  0.  Otherwise  to  each  8k  of  a  sequence  of 
constants  approaching  zero,  there  would  correspond  a  point 
(x)k  in  the  region  As>c  which  would  belong  to  no  solution  in  Pg. 
To  each  (x)k  there  would  correspond  a  point  (a)k  in  A  satisfying 
the  inequalities 


x  —  a 


< 


with  the  values  (x)k,  and  the  points  (a)k  would  have  a  point  of 
condensation  a  in  A,  which  would  also  be  a  point  of  condensation 
for  the  sequence  (x)k,  since  A  is  finite  and  closed  when  P  is  so. 
But  by  the  original  theorem  of  §  1,  again,  it  is  known  that  a 
neighborhood  as  of  a  can  be  chosen  in  which  every  point  x  has 
associated  with  it  a  solution  (x;  y)  in  pe,  where  p(a;b)  is  the 
point  of  P  having  the  projection  a.  Consequently  the  existence 
of  the  sequence  (x)k  is  contradicted. 

If  now  the  region  Pe  is  so  restricted  that  the  functional  de- 
terminant D(x;  y)  remains  different  from  zero  throughout  it, 
then  the  original  theorem  of  §  1  can  be  applied  to  show  that  the 
functions  y(xi,  x2,  •  •  • ,  xn)  are  continuous  at  any  point  of  the 
region  A&  and  possess  as  many  continuous  derivatives  as  are  pos- 
sessed by  the  functions  f(x;  y). 

§  5.    THE  UNIQUE  SHEET  OF  SOLUTIONS  ASSOCIATED  WITH  AN 
INITIAL  SOLUTION 

The  points  of  the  space  (x;  y)  may  be  divided  into  two  classes, 
ordinary  points  and  exceptional  points,  with  respect  to  the  func- 
tions /.  An  ordinary  point  is  one  at  which  the  first  and  third 
hypotheses  of  the  theorem  of  §  1  are  postulated,  that  is,  one  near 
which  the  functions/  and  their  first  derivatives  with  respect  to  y 
are  continuous  and  the  functional  determinant  D  =  d(/i,/2,  •  •  •, 


22  THE   PRINCETON   COLLOQUIUM. 

fn)/d(y\,  yz,  '  •  •,  yn)  is  different  from  zero.  An  exceptional  point 
is  one  at  which  some  of  these  conditions  are  not  fulfilled  or  are 
not  presupposed. 

A  sheet  of  points  in  the  (ra  -f-  w)-dimensional  space  (x;  y} 
may  be  defined  as  a  point  set  S  with  the  property  that  for  any 
point  p(a;b)  belonging  to  the  set  a  neighborhood  pf  can  always 
be  found  such  that  no  two  points  of  S  in  pf  have  the  same  pro- 
jection x.  In  other  words,  the  variables  y  are  single-valued 
functions  y(x\,  ar2,  •••,  xm)  in  the  neighborhood  of  the  point  p, 
for  points  of  the  sheet. 

If  for  any  neighborhood  6,  of  the  kind  just  described,  a  region 
as  (5^  «)  can  be  found  in  which  every  point  x  belongs  to  a 
point  of  S  in  pt,  then  p  is  said  to  be  an  interior  point  of  the  sheet  S. 

A  boundary  point  is  a  limit  point  of  points  of  the  sheet,  which 
is  not  itself  an  interior  point  and  may  not  even  belong  to  S. 

A  sheet  is  said  to  be  connected  if  every  pair  (x';yf),  (x"\y") 
of  its  interior  points  can  be  joined  by  a  continuous  curve 

*=*(*),       y  =  y(t)  (t'^t^t"), 

consisting  entirely  of  interior  points  of  the  sheet. 

In  the  following  pages  it  is  always  to  be  understood  that  the 
sheets  considered  are  continuous  and  have  continuous  first 
derivatives,  or  in  other  words  at  any  interior  point  of  one  of  them 
the  functions  ^(a-i,  z2,  •  • -,  xm}  mentioned  above  have  these 
properties.  A  sheet  will  be  said  to  become  infinite  near  a  point 
x'  if  x'  is  the  limit  of  the  projections  of  a  sequence  of  points  (x;  y) 
of  the  sheet  for  which  one  at  least  of  the  variables  y  approaches 
infinity. 

With  the  preceding  agreements  as  to  nomenclature  in  mind, 
it  is  possible  to  prove  the  following  theorem: 

//  a  point  p(a;  6)  is  an  ordinary  point  for  the  functions  f  and 
satisfies  the  equations  f  =  0,  then  there  passes  through  p  one  and 
only  one  connected  sheet  of  solutions  of  these  equations,  with  the 
properties : 

1)  all  points  of  the  sheet  are  ordinary  points  of  the  functions  f; 

2)  all  points  are  interior  points', 


FUNDAMENTAL  EXISTENCE  THEOREMS.  23 

3)  the  only  boundary  points  of  the  sheet  are  exceptional  points  for 
the  system  f. 
The  set  of  points 

[*i,  x2,  •  •  •,  xm;  y\(xi,  x2)  •  •  -,  xm),     •  •  •,    yn(xi,  z2,  •  -  •,  xm}} 

defined  over  the  region  a&  by  the  principal  theorem  of  §  1,  is  a 
sheet  *Si  of  solutions  of  the  equations/  =  0  which  satisfies  all  the 
requirements  of  the  theorem  just  stated  except  possibly  the  last. 
Its  points  are  all  interior  points  since  the  region  as  is  defined  by 
inequalities  only.  If  any  boundary  point  p'(af;  b')  of  S\  is  an 
ordinary  point  of  the  functions  /  it  must  satisfy  the  equations 
/  =  0,  since  the  /'s  are  continuous  and  p'  is  a  limit  point  of  points 
on  Si.  Consequently  the  theorem  of  §  1  can  be  applied  in  the 
neighborhood  of  p',  and  the  sheet  S'  so  determined  near  p' 
forms  with  Si  a  new  set  S2.  This  process  may  be  repeated  any 
number  of  times,  and  the  totality  of  points  which  can  be  attained 
by  a  finite  number  of  such  extensions,  constitutes  the  sheet  S 
required  in  the  theorem. 

The  set  of  points  S  so  determined  constitutes  a  sheet,  since 
any  point  q  of  it  is  an  ordinary  point  and  a  solution  of  the  equa- 
tions /  =  0,  and  according  to  the  theorem  of  §  1  the  solutions  of 
these  equations  in  the  neighborhood  of  q  have  the  property  which 
is  characteristic  of  a  sheet.  From  the  manner  of  its  construction 
the  sheet  is  evidently  connected  and  consists  entirely  of  interior 
points.  If  any  boundary  point  q  of  S  were  an  ordinary  point  of 
the  functions  /,  the  sheet  could  be  extended  to  include  q  as  an 
interior  point  by  the  process  described  in  the  preceding  paragraph. 

There  could  not  be  a  second  sheet  S  containing  a  point  TT 
not  in  S  and  having  the  properties  stated  in  the  theorem.  For 
there  would  in  that  case  be  a  continuous  curve 

x  =  x(t),    y  =  y(t)  (h^t^  #2) 

in  S  joining  p  with  TT  and  consisting  entirely  of  ordinary  points. 
In  a  neighborhood  of  t  =  ti  all  of  the  points  defined  on  the  curve 
would  also  be  points  of  S,  since  the  solutions  of  the  equations 


24  THE   PRINCETON   COLLOQUIUM. 

/  =  0  near  the  initial  point  p  of  the  curve  are  all  in  S.  The  values 
of  t  defining  points  on  the  curve  and  in  S  would  therefore  have 
an  upper  bound  T  ^  i*  such  that  T  would  define  on  the  curve  a 
boundary  point  of  S.  But  this  is  impossible  since  all  of  the 
points  of  the  curve  are  ordinary  points. 

If  the  functions  /  are  known  to  be  continuous  and  to  have  con- 
tinuous derivatives  in  a  region  R,  then  it  follows  readily  from 
what  precedes  that  through  any  ordinary  solution  of  the 
equations/  =  0  interior  to  R  there  passes  one  and  only  one  sheet 
of  solutions  having  the  property  that  the  only  boundary  points 
of  the  sheet  are  boundary  points  of  R,  or  interior  points  of  R  at 
which  the  functional  determinant  vanishes.  If  R  is  finite  and 
closed  and  consists  entirely  of  ordinary  points  for  the  functions/, 
then  there  can  not  be  more  than  a  finite  number  of  points  of 
the  sheet  on  any  ordinate  x.  Otherwise  the  points  common  to 
the  ordinate  and  the  sheet  would  have  a  point  of  condensation  p, 
also  in  R.  Since  p  is  an  ordinary  point  there  can  be  at  most  one 
solution  of  the  equations  in  a  properly  chosen  neighborhood  pe. 

It  is  interesting  to  determine  a  criterion  which  shall  characterize 
a  sheet  which  is  at  most  single-valued  on  any  ordinate.  Such  a 
criterion  is  derived  in  §  7  in  connection  with  a  theorem  due 
originally  to  Schoenflies,  and  afterwards  proved  by  Osgood. 
The  proof  of  it  involves  the  auxiliary  notions  described  in  §  6 
and  the  following  corollaries  to  the  preceding  theorem: 

//  the  initial  point  of  a  continuous  arc 

(C*)  Xi  =  Xi(t)          (i  =  1,  2,  •  •  •,  m\  t'  <t<  t") 

in  the  x-space  is  the  projection  of  a  solution  p'(x';  y')  of  the  equations 
/  =  0  which  is  an  ordinary  point  for  the  functions  f,  then  there  is 
associated  with  the  arc  Cx  one  and  only  one  continuous  curve 

(Cxv)      Xi=Xi(t),     ya=y«(0         (i=l,  2,  •••,  m\  a=l,  2,  ••-,  n) 

passing  through  (xf;  y')  for  t  =  t',  with  the  properties: 
1)  all  of  its  points  are  solutions  of  the  equations  /  =  0  and  or- 
dinary points  of  the  functions  /; 


FUNDAMENTAL  EXISTENCE  THEOREMS.  25 

2)  it  is  defined  either  over  the  whole  interval  t'  ^  t  ^  t",  or  else 
on  an  interval  t'  ^  t  <  r  ( ^  t")  such  that  as  t  approaches  r 
the  only  limit  points  of  the  curve  Cxy  are  at  infinity  or  are  excep- 
tional points  of  the  functions  f. 

The  truth  of  this  statement  is  readily  deduced  from  the  con- 
siderations which  precede,  or  by  the  following  argument.  The 
fundamental  theorem  of  §  1  can  be  applied  at  the  point  (.r';  y'}. 
If  the  arc  Cx  is  entirely  within  the  region  xs'  then  the  existence 
and  uniqueness  of  the  curve  Cxy  is  evident.  In  any  case  there 
will  be  some  intervals  t'  ^  t  ^  ti'm  which  curves  Cxy  are  defined 
having  all  the  properties  described  in  the  theorem  except  possibly 
2).  Suppose  that  T  is  the  upper  bound  of  the  end  values  ti 
for  such  intervals.  Then  there  is  a  curve  Cxy  well  defined  in 
the  interval  t'  ^  t  <  r,  and  no  limit  point  (a.  ;  j8)  of  the  curve 
as  t  approaches  T  can  be  a  finite  ordinary  point  for  the  func- 
tions /.  For  if  (a.  ;  {$)  were  such  a  point,  it  would  also  satisfy 
the  equations/  =  0,  on  account  of  the  continuity  of  the  functions 
/,  and  the  theorem  of  §  1  could  again  be  applied  at  (a.  ;  j8).  A 
curve  Cxy  with  all  the  properties  of  the  theorem,  except  possibly 
2),  could  then  be  defined  over  an  interval  including  the  interval 
t'  ^  t  <  r  in  its  interior,  which  contradicts  the  assumption  that 
r  is  the  upper  bound  of  such  intervals. 

There  could  not  be  two  curves  Cxy  associated  with  the  projection 
Cx,  having  the  properties  described  in  the  theorem,  and  having 
distinct  points  (x;  y'}  and  (x;  y"}  corresponding  to  the  same 
value  t<>.  For  if  so,  there  would  be  an  interval  t%  <  t  ^  t2  in 
which  the  curves  would  be  distinct  while  at  t  =  t3  they  coincide. 
This  is,  however,  impossible  since  in  a  neighborhood  of  the  point 
corresponding  to  t3  there  can  be  but  one  solution  of  the  equations 
/  =  0  corresponding  to  a  given  set  of  values  x. 

Suppose  that  a  continuum  X  of  points  (x\,  x<i,  •  •  •,  xm)  contains 
no  projection  of  a  boundary  point  of  a  sheet  S  of  solutions  of  the 
equations  f  =  0,  and  no  point  near  which  the  sheet  becomes  infinite. 
Then  if  X  contains  the  projection  of  a  point  on  the  sheet  every  other 
point  of  X  will  also  be  such  a  projection.  On  the  other  hand,  if  X 


26  THE   PRINCETON  COLLOQUIUM. 

contains  a  point  which  is  not  a  projection  of  any  point  of  the  sheet, 
then  no  point  of  X  can  be  a  projection  of  a  point  of  S. 

These  statements  follow  readily  with  the  help  of  the  last 
theorem.  For  suppose  that  X  contains  the  projection  x'  of 
an  interior  point  (x1;  y'}  of  a  sheet  of  solutions  of  the  equations 
/  =  0,  and  let  x"  be  any  other  point  of  X.  Since  X  is  a  continuum 
there  exists  a  continuous  arc  Cx  entirely  interior  to  X  joining 
x'  and  x",  and  the  corresponding  continuation  curve  Cxv  must  be 
defined  over  the  whole  of  the  arc  Cx.  Hence  x"  is  also  the  pro- 
jection of  a  point  of  the  sheet  of  solutions  through  (x';  y').  The 
rest  of  the  theorem  follows  at  once. 

//  the  curve  Cxv  in  the  last  theorem  but  one  is  defined  over  the 
whole  arc  Cx,  and  has  initial  and  end  points  p'  and  p",  respectively, 
then  there  always  exists  a  positive  constant  p  such  that  any  curve  Tx 
lying  in  the  p-neighborhood  of  the  curve  Cx  and  joining  x'  to  x", 
has  a  unique  continuation  curve  Txy  also  joining  p'  and  p". 

The  curve 
(F*)  x  =  &(«)       (i=  1,2,  --,m;  u'  <^u  ^  u"} 

is  said  to  lie  in  the  p-neighborhood  of  Cx  if  there  exists  a  continuous 

function 

(14)  t  =  t(u)  (u'  ^  u  ^  u") 

taking  the  values  t ',  t"  at  the  ends  of  the  w-interval,  and  such 
that  the  point  a  on  Tx,  defined  by  any  value  of  u,  lies  in  the  neigh- 
borhood ap  of  the  corresponding  point  a  of  Cx  determined  by 
the  relation  (14). 

It  is  possible  to  choose  two  constants,  e  and  S  ^  «,  so  that  the 
neighborhoods  pt  and  as  have  the  properties  described  in  the 
theorem  of  §  1  uniformly  for  every  point  p(a,  b)  on  the  arc  Cxy. 
If  not,  there  would  be  a  sequence  of  points  pk  on  Cxy  with  a  limit 
point  TT,  for  which  the  largest  possible  constants  e*  have  the 
limit  zero.  But  for  the  point  TT  there  is  an  effective  constant 
e  >  0,  and  the  constants  e*  could  not  therefore  decrease  indefi- 
nitely in  size.  A  similar  argument  shows  the  existence  of  the 
constant  6. 


FUNDAMENTAL  EXISTENCE  THEOREMS. 


27 


Suppose  now  that  the  interval  u'  ^  u  ^  u"  is  divided  by 
values  Uk  (k  =  1,  2,  •  •  •,  v)  into  sub-intervals  so  small  that  the 
points  of  any  arc  ak-i<Xk,  corresponding  on  Tx  to  the  values 
Uk-i  ^  u  ^  Uk,  all  lie  in  the  ^5-neighborhood  of  the  point  ctk-i, 
and  further  so  small  that  the  same  is  true  with  respect  to  the 
point  dk-i  of  the  arc  dk-idk  of  Cx  corresponding  to  cck-i<Xk  by 
means  of  the  relation  (14).  The  constant  p  is  supposed  to  have 


FIG.  l. 

been  chosen  equal  to  %8,  so  that  the  curve  F  lies  in  the  |8- 
neighborhood  of  C.  Then  the  four-sided  closed  curve  formed  by 
the  two  straight  lines  dk-ictk-i  and  dkctk,  and  the  two  arcs  dk-idk 
and  oik-\ak,  lies  entirely  within  the  6-neighborhood  of  the  point 
cik-i.  The  two  continuation  curves  in  the  xy-space,  starting  with 
the  point  pk-i  on  Cxv  and  having  as  projections  the  arcs  ak-idkctk 
and  dk-iak-icxk,  respectively,  lead  to  the  same  point  irk  corre- 
sponding to  the  point  &k  in  the  z-space. 

It  is  possible  to  argue,  then,  that  the  point  TTI  on  the  continu- 
ation curve  of  the  arc  a'ai  is  the  same  as  that  of  the  continuation 
curve  for  a'aiai,  since  the  arcs  a'ai  and  a'a\a\  lie  entirely  within 
the  5-neighborhood  of  the  point  a\.  Similarly,  the  point  7r2 
for  the  arc  a'a2  is  the  same  as  that  for  the  continuation  curve 
along  d'dzoiz-  And  finally  the  point  IT"  must  coincide  with  p", 
provided  always  that  the  initial  points  IT'  and  p'  of  the  con- 
tinuation curves  are  the  same. 


28  THE  PRINCETON   COLLOQUIUM. 

In  particular  if  the  curve  Cxy  is  defined  over  the  whole  arc  Cx, 
as  described  above,  then  there  exists  a  polygon  in  the  x-space  joining 
a'  and  a"  in  the  p-neighborhood  of  Cx,  and  along  ivhich  there  is  a 
continuation  curve  in  S  also  joining  p'  and  p".  The  polygon  can 
be  so  chosen  that  no  two  adjacent  sides  have  more  than  an  end  point 
in  common. 

To  show  this,  let  the  interval  t'  ^  t  ^  t"  be  divided  in  any 
way  by  means  of  points  of  division  t',  t2,  t3,  •  •  •  ,  tv,  t",  and  let 
the  corresponding  points  on  the  curve  Cxy  be  (x';  y'},  (£";  17"), 
•",  (S00;  -n(v)),  (x"\  */")•  The  straight  line  £<*)£<*+i>  has  the 
equations 


Since  the  functions  defining  Cx  are  continuous,  and  therefore 
uniformly  continuous,  in  t'  ^  t  ^  t",  it  is  possible  to  take  the 
points  of  division  t',  t%,  t3,  •  •  •  ,  tv,  t"  so  close  together  that  the 
differences  x  —  £(fc),  for  any  point  x  on  the  arc  £W£(fc+1>  of  Cx,  are 
uniformly  less  than  an  arbitrarily  assigned  positive  constant  5; 
and  the  preceding  theorem  shows  that  the  curve  Cxy  and  the 
Continuation  curve  along  the  polygon  both  lead  from  p'  to  p". 
If  the  sides  £<*>£<*+»  and  £<w-«£<*+2)  have  more  than  the  point 
£(k+D  m  common,  then  one  of  the  two  would  be  included  entirely 
within  the  other,  and  the  continuation  curve  along  £<*)£(*r+2 
would  have  the  same  end  points  as  that  along  the  two  successive 
sides.  Therefore,  by  replacing  adjacent  sides  by  a  single  one 
whenever  the  two  have  more  than  one  end  point  in  common,  a 
polygon  as  described  in  the  theorem  can  be  found. 

§  6.    AUXILIARY  THEOREMS  AND  DEFINITIONS 

In  this  section  it  is  proposed  to  record  some  theorems  which 
will  be  of  service  later,  especially  in  the  proofs  of  the  theorems  of 
§  7.  In  the  first  place  let  it  be  agreed  that  a  regular  curve  in 
the  plane  shall  mean  one  which  is  continuous  and  has  a  well- 
defined  tangent  at  all  except  possibly  a  finite  number  of  points, 


FUNDAMENTAL   EXISTENCE   THEOREMS.  29 

at  each  of  which,  however,  the  slope  of  the  tangent  approaches 
definite  limits  as  the  point  is  approached  from  either  side. 
Analytically  this  means  that  the  functions 

x  =  x(t),        y  =  y(t)  (?  £t£  t") 

defining  a  regular  curve  are  continuous  in  the  whole  interval 
t'  ^  t  ^  t",  that  they  are  differentiate  and  satisfy  the  inequality 

(15)  (dxfdt)2  +  (dy/dt)2  4=  0 

at  all  except  possibly  a  finite  number  of  values  of  t.  At  an  ex- 
ceptional value  t  =  T,  where  the  derivatives  are  not  well  defined 
or  where  the  expression  (15)  vanishes,  the  angle  <p  defined  by  the 
equations 

dx/dt  .  dy/dt 

+  (dy/dt)2'        n<P'     ^(dx/dt)2  +  (dy/dt)2 


cos<p= 


has  nevertheless  a  unique  limit  as  t  approaches  T  on  the  right, 
and  a  unique  limit  as  t  approaches  T  on  the  left.  These  two 
limits  are  not  necessarily  the  same. 

It  is  known  that  a  simply  closed  regular  curve  C  in  an  xy- 
plane  divides  the  plane  into  two  continua,  an  exterior  and  a 
finite  interior.*  Any  two  interior  points  can  be  joined  by  a 
regular  curve  every  point  of  which  is  an  interior  point,  and  a 
similar  statement  holds  for  exterior  points.  Any  continuous 
curve  joining  an  interior  and  an  exterior  point  must  have  on  it 
at  least  one  point  of  the  curve  C,  and  any  point  p  on  C  can  be 
joined  with  an  interior  point  by  a  regular  curve  which  has  in 
common  with  C  only  the  point  p. 

The  interior  of  a  simply  closed  regular  curve 

x  =  x(t),        y  =  y(t)  (t'£t£  t") 

can  be  divided  by  a  finite  number  of  segments  of  straight  lines  into 

*  See  for  example  Osgood,  Lehrbuch  der  Funktionentheorie,  Chapter  V, 
§§  4-6;  Bliss,  "A  proof  of  the  fundamental  theorem  of  analysis  situs,"  Bulletin 
of  the  American  Mathematical  Society,  vol.  12  (1906),  page  336. 


30  THE   PRINCETON   COLLOQUIUM. 

regions  each  of  which  has  a  maximum  diameter  less  than  an  ar- 
bitrarily assigned  positive  constant  e.* 

Let  the  maximum  and  minimum  values  of  y  in  the  interval 
t'  ^  t  ^  t"  be  7/1  and  t/2,  and  let  pi  and  pz  be  two  points  of  C 
at  which  y  has  these  values.  It  is  desired  to  show  that  there  is 
a  segment  p'p"  of  the  horizontal  line  y  =  (y\-\-  yz)/2  which 
forms  with  C  two  simply  closed  regular  curves,  p'p\p"p'  and 
p'pzp"p',  each  containing  one  of  the  points  p\  and  p%. 

The  points  p\  and  p2  can  be  joined  by  a  regular  curve  D  which, 
except  at  its  end  points,  is  interior  to  C.  Two  arcs  of  D  adjoining 
pi  and  pz,  can  be  marked  off  in  such  a  way  that  they  do  not  cut 
the  line  y  =  (yi~\-  yz)/2.  The  remaining  arc  D'  of  D  is  entirely 
interior  to  C  and  can  be  replaced  by  a  continuous  polygon  D" 
with  a  finite  number  of  sides,  having  the  same  end  points  and 
consisting  also  of  interior  points  of  C  only.  Any  side  of  D" 
which  has  an  end  point  in  common  with  the  line  y  =  (y\-\-  t/2)/2 
may  be  rotated  slightly  about  its  other  end  point,  and  in  this 
way  it  may  be  brought  about  that  D"  has  only  interior  points 
of  its  sides  on  the  line  y  =  (y\  +  ?/2)/2,  and  actually  crosses  the 
line  wherever  they  have  a  point  in  common. 

The  polygon  D"  must  intersect  y  =  (y\-\-  yz)/2  at  least  once, 
say  at  a  point  p,  since  one  end  point  of  D"  is  above  and  the  other 
below  this  line.  There  will  be  a  segment  p'p"  of  y=  (2/i+2/2)/2, 
containing  p  and  such  that  pf  and  p"  are  on  the  curve  C  while 
every  other  point  of  the  segment  is  interior  to  C.  There  can  be 
only  a  finite  number  of  such  segments  p'p"  containing  points 
of  D",  since  D"  has  at  most  a  finite  number  of  intersections  with 
the  horizontal  line.  There  must  be  at  least  one  segment  on 
which  D"  has  an  odd  number  of  intersection  points,  since  other- 
wise both  end  points  of  D"  would  be  on  the  same  side  of 
y  —  (y\  +  2te)/2.  If  p'p"  is  such  a  segment,  then  it  forms  with 
C  two  simply  closed  regular  curves  p'p\p"p'  and  p'p2p"p', 
one  of  which  contains  pi  and  the  other  p2.  For  after  its  last 
intersection  with  p'p"  the  polygon  D"  and  hence  p2  is  entirely 
exterior  to  the  curve  p'p\p"p'. 

*  For  a  similar  theorem  see  Osgood,  loc.  cit.,  Chapter  V,  §  9. 


FUNDAMENTAL  EXISTENCE  THEOREMS.  31 

For  the  moment  that  part  of  a  curve  which  does  not  lie  in  a 
horizontal  line  may  be  called  the  effective  arc  of  the  curve,  in 
view  of  the  fact  that  the  altitude  of  the  curve  can  not  be  more 
than  one  half  the  length  of  this  so-called  effective  part.  If  the 
altitude  of  any  curve  is  ^  e,  the  effective  length  of  either  of  its 
two  parts  after  subdivision  by  a  horizontal  line  segment,  as 
described  above,  will  be  ^  L  —  e,  where  L  is  its  effective  length. 

If  the  altitude  y\  —  y%  of  C  is  greater  than  e,  then  the  effective 
arc  of  either  p'pip"p'  or  pfp2p"p'  will  be  greater  in  length  than  «, 
and  the  effective  length  of  each  will  also  be  less  than  L  —  e, 
where  L  is  the  perimeter  of  C.  If  the  curve  prp\p"p',  for  example, 
has  still  an  altitude  greater  than  e,  it  may  be  subdivided  by  a 
horizontal  segment  as  before,  and  the  effective  arcs  of  the  two 
new  curves  so  found  will  be  less  than  L  —  2e.  By  a  continu- 
ation of  this  process  the  interior  of  C  will  be  subdivided  finally 
by  curves  whose  effective  lengths  are  less  than  2e  and  whose 
altitudes  are  therefore  less  than  e. 

In  a  similar  manner  the  regions  so  formed  may  be  subdivided 
by  vertical  segments  into  others  whose  breadths  are  less  than  e, 
and  the  theorem  follows  at  once. 

A  set  of  points  in  an  #i£2-plane  is  connected  if  any  two  of  its 
points  can  be  joined  by  a  continuous  arc  whose  points  all  belong 
to  the  set,  and  it  is  further  said  to  be  simply  connected  if  every 
simply  closed  regular  curve  in  it  has  an  interior  which  also 
consists  only  of  points  of  the  set. 

It  is  more  difficult  to  set  down  a  satisfactory  definition  of 
simple  connectivity  for  sets  of  points  in  an  m-dimensional  space. 
In  the  following  section  of  these  lectures,  however,  a  special  type 
of  simple  connectivity  is  needed  which  may  be  defined  by  means 
of  some  simple  auxiliary  conceptions. 

A  normal  subspace  of  two  dimensions  in  a  region  X  of  points 
(xi,  Xz,  •  •  • ,  Xm)  is  a  totality  of  points  defined  by  equations  of  the 
form 

Xi  =  <pi(ui,  Uz)  (i  =  1,  2,  •  •  •,  m), 

where 


32  THE   PRINCETON   COLLOQUIUM. 

1)  the  values  (MI,  1/2)  range  over  a  simply  connected  region  U; 

2)  no  two  distinct  sets  of  values  u  define  the  same  point  x; 

3)  the  functions  <p  are  continuous  and  have  continuous  first 
derivatives  in  U', 

4)  the  determinants  of  the  second  order  of  the  matrix  of 
derivatives  ||d^>,-/dwjfc||   (i  =  1,  2,   •  •  •,  m;  k  =  1,  2)  do  not  all 
vanish  simultaneously  at  any  point  of  U. 

A  simply  connected  region  in  two  dimensions  is  defined  above, 
and  a  connected  region  A"  in  a  space  of  points  (xit  x2,  •  •  • ,  xm) 
has  a  definition  quite  similar  to  that  for  two  dimensions.  In 
order  to  specify  conveniently  the  properties  of  a  region  X  which 
is  simply  connected,  the  term  elementary  curve  will  also  be  used. 
By  an  elementary  curve  in  X  is  meant  a  simply  closed  continuous 
curve  which  either  lies  in  a  normal  subspace  of  two  dimensions 
entirely  in  the  interior  of  A',  or  else  is  such  that  in  every  neighbor- 
hood of  it  there  is  a  simply  closed  continuous  curve  having  this 
property.  It  is  thus  seen  that  while  an  elementary  curve  may 
not  itself  be  imbedded  in  one  of  the  two-dimensional  normal  sub- 
spaces  interior  to  X,  it  can  nevertheless  be  approximated  as 
closely  as  may  be  desired  by  one  which  does.  The  word  neighbor- 
hood is  here  used  in  the  sense  described  in  connection  with  the 
fourth  theorem  of  §  5  (see  page  26) . 

If  a  region  X  is  connected,  then  any  simply  closed  continuous 
curve  in  its  interior  may  be  developed  into  two  such  curves  by 
an  auxiliary  arc  joining  two  of  its  points,  and  the  process  of 
development  may  be  continued  on  the  two  arcs  so  formed. 

//  a  region  X  is  such  that  any  simply  closed  continuous  curve  in 
its  interior  is  an  elementary  curve,  or  may  be  developed  into  a 
number  of  elementary  curves  by  means  of  auxiliary  arcs,  as  just 
described,  then  X  is  said  to  be  simply  connected* 

*  For  a  discussion  of  the  connectivity  of  higher  spaces,  see  Picard  and  Simart, 
Th6orie  des  Fonctions  alg^briques  de  deux  Variables  independantes,  Chapitre 
II,  in  particular  §§  11  ff.  If  every  simply  closed  continuous  curve  interior 
to  R  lies  in  a  normal  subspace  of  two  dimensions  interior  to  R,  one  sees  intu- 
itively that  a  second  neighboring  subspace  of  the  same  kind  can  be  passed 
through  the  curve.  The  closed  two-dimensional  subspace  so  formed  is 


FUNDAMENTAL   EXISTENCE   THEOREMS.  33 

§  7.    A   CRITERION  THAT  A  SHEET  OF  SOLUTIONS  BE  SINGLE- 

VALUED 

Consider  in  the  first  place  a  set  of  equations 
(16)  fa(xi,  x2;  yi,  y2,  •  -  -,  yn)  =  0      (a  =  1,  2,  •  •  •,  ri) 

in  which  there  are  but  two  independent  variables  x. 

If  a  connected  sheet  S  of  solutions  of  equations  (16)  consists  only 
of  ordinary  points  of  the  functions  f,  and  furthermore  has  a  simply 
connected  projection  X  in  the  XiX^-plane  such  that  no  interior 
point  of  X  is  either  a  point  where  S  becomes  infinite  or  the  pro- 
jection of  a  boundary  point  of  S,  then  the  sheet  S  is  single-valued 
over  the  interior  of  X. 

Suppose,  in  contradiction  to  the  theorem,  that  over  any 
interior  point  of  X  there  were  two  points,  p'  and  p",  of  the  sheet. 
Since  S  is  connected  there  would  be  a  continuous  curve 


(Cxy)    xi  =  xi(t),    x2  =  x2(f),    ya  =  ya(t) 

(?  £t  £  i"\  a=  1,  2,  .--,71) 

consisting  entirely  of  interior  points  of  the  sheet  and  joining  p' 
with  p"  in  the  space  (x;  y).     The  projection 

(C.)  an  =  antf),        x,  =  x2(t)  (f  £  t£  t") 


of  this  curve  would  necessarily  be  a  closed  curve  in  the 
and  by  the  second  theorem  of  §  5  the  arc  Cxy  is  the  only  one 
associated  with  Cx  in  the  sheet  S  and  having  the  initial  point  p'. 
The  curve  Cx  may  be  simply  closed  and  regular;  but  if  it  is 
not,  there  will  nevertheless  be  a  curve  in  the  region  X  having 
these  properties,  and  for  which  the  continuation  curve  analogous 
to  Cxy  is  not  closed.  For,  in  the  first  place,  from  §  5  it  is  seen 
that  the  curve  Cx  may  be  supposed  to  be  a  polygon  no  two  ad- 
jacent sides  of  which  have  more  than  an  end  point  in  common, 
provided  that  it  is  desired  only  to  secure  a  continuous  curve  in 

separated  into  two  parts  by  the  curve,  and  hence  the  number  which  Picard 
and  Simart  designate  by  p\  is  equal  to  unity  for  a  simply  connected  region  of 
the  kind  denned  in  the  text  above. 
4 


34  THE   PRINCETON   COLLOQUIUM. 

the  sheet  passing  from  p'  to  p".  Let  the  corners  of  this  polygon 
in  the  a>plane  be  denoted  by  £1,  £2,  •  •  • ,  £„,  where  £  is  a  symbol  for 
a  point  (x\,  x2).  The  side  £,,£1  touches  £i£2  at  its  end  point  £1, 
and  it  can  be  argued  therefore  that  there  will  be  some  first  side 
£A£A+I  which  touches  some  one  of  the  preceding  sides  elsewhere 
than  at  its  initial  point  £A.  Let  the  side  so  touched  by  £A£A+I 
be  £«£«+!»  where  K  +  1  is  necessarily  less  than  \,  and  let  the  first 
point  of  £A£A+I  which  lies  on  £,£*+!  be  £.  If  the  portion  of  the 
curve  Cxy  which  corresponds  to  the  polygon 

(17)  £,    £.+i,    £.+2,     ••-,    £A,    £ 

is  not  closed,  then  the  polygon  (17)  itself  is  a  simply  closed  curve 
in  X  of  the  kind  desired  above,  that  is,  one  along  which  there 
exists  a  continuation  curve  in  the  xy-space  whose  end  points 
are  different. 

If  the  portion  of  Cxy  which  corresponds  to  (17)  is  closed,  then 
that  part  of  Cxy  which  belongs  to  the  polygon 

(18)  £1,  £2,  •••,  £«,  £,  £A+I,  •**,  £„,  £1 

is  also  continuous  and  leads  from  p'  to  p".  Since  K  +  1  <  X 
the  side  £(C+i£)t+2  at  least  is  missing  in  (18),  and  the  number  of 
sides  is  at  least  one  less  than  that  of  the  original  polygon.  By 
an  alteration  of  the  kind  suggested  in  the  proof  of  the  last  theorem 
of  §  5,  which  also  reduces  the  number  of  sides,  it  can  be  brought 
about,  if  not  already  true,  that  the  polygon  (18)  still  has  no 
two  adjacent  sides  with  more  than  an  end  point  in  common. 

By  continuing  this  process  one  must  come  at  some  stage  to  a 
simply  closed  regular  curve  in  the  ar-plane  with  a  corresponding 
continuation  curve  in  the  xy-space  which  is  not  closed.  In  order 
not  to  complicate  the  notation  too  much  it  may  be  supposed 
that  the  curve  Cx  itself  is  such  a  curve.  Every  point  of  Cx  is 
an  interior  point  of  the  region  X  since  the  corresponding  point 
of  Cxy  is  an  interior  point  of  the  sheet  S.  The  interior  of  Cx 
is  therefore  also  composed  entirely  of  interior  points  of  X,  since 
X  is  simply  connected.  If  the  interior  of  Cx  is  subdivided  into 


FUNDAMENTAL  EXISTENCE  THEOREMS.  35 

two  parts  by  a  segment  of  a  straight  line,  as  described  in  the  pre- 
ceding section,  the  dividing  segment  will  also  have  a  continu- 
ation curve  on  the  sheet  S  throughout  its  entire  length,  by  the 
second  theorem  of  §  5.  For  its  initial  point  on  the  curve  Cx 
corresponds  to  an  interior  point  of  the  sheet  S  and,  by  the  hy- 
pothesis of  the  theorem  which  is  to  be  proved,  none  of  its  points 
can  be  a  point  where  S  becomes  infinite  or  can  correspond  to  a 
boundary  point  of  S.  Hence  one  of  the  simply  closed  curves 
formed  by  the  curve  Cx  and  the  dividing  segment  is  a  curve 
retaining  the  property  that  it  has  a  continuation  curve  on 
the  sheet  S  which  is  not  closed.  Suppose  that  Cxr  is  this  curve. 
By  continuing  the  process  a  sequence  of  curves  { Cx(k) } ,  with 
diameters  approaching  zero,  can  be  found,  each  lying  in  the 
interior  of  Cx  and  having  an  unclosed  continuation  curve  CXV(K* 
onS. 

If  a  point  p(k)  is  selected  arbitrarily  on  the  curve  Cxy(k),  the 
sequence  {p(k) }  (k=  1, 2,  •  •  • ,  oo )  will  have  a  finite  point  of  conden- 
sation 7r(a;  j8)  in  the  xy-space  which  is  an  interior  point  of  the 
sheet  S.  For  the  projections  x(k)  of  the  points  p(k)  all  lie  in  the  in- 
terior of  Cx  and  hence  must  have  a  point  of  condensation  a.  Fur- 
thermore the  points  of  the  sequence  p(k)  whose  projections  are  in  the 
neighborhood  of  a  can  not  become  infinite  or  approach  a  boundary 
point  of  the  sheet,  since  a.  is  interior  to  X.  They  must  therefore 
have  at  least  one  limit  point  ir  which  is  an  interior  point  of  the 
sheet,  and  with  which  there  are  associated  two  neighborhoods 
Tre  and  as  by  the  principal  theorem  of  §  1.  Some  of  the  points 
p(®  lie  in  irf,  and  have  corresponding  curves  Cx(k)  in  as.  For 
such  points  the  continuation  curves  Cxy(k)  also  lie  in  ire  and  can 
not  be  unclosed,  since  to  any  point  x  in  as  there  corresponds  in 
TT€  at  most  one  solution  of  the  equations  /  =  0.  The  original 
assumption  that  S  is  multiple-valued  in  the  interior  of  X  is 
therefore  contradicted. 

The  theorem  remains  true  for  any  system  of  equations  of  the  form 

(19)    /.(an,  x2,  ••-,*„;  yi,  y2,  •••,  #»)  =  0     (a  =  1,  2,  •  •  -,  n). 


36  THE   PRINCETON   COLLOQUIUM. 

In  this  case  the  curves  Cxv  and  Cx  have  equations 

(CW)  *i  =    *t(0,  2/a   =    2/a(0 

(i=  1,2,  •  •  •,  m;  a  =  1,  2,  •  •  •,  n;  t'  ^  t  ^  O, 
(Cx)  .T,  =  *,-(*), 

and  the  question  asked  in  the  proof  of  the  theorem  just  stated 
is  whether  or  not  the  latter  curve  may  be  closed  while  the  former 
has  distinct  end  points. 

It  is  a  part  of  the  hypothesis  of  the  theorem  that  the  region 
X  is  simply  connected  according  to  the  definition  of  the  preceding 
section;  and,  according  to  the  arguments  made  in  the  paragraphs 
above,  the  curve  Cx  may  be  supposed  a  simply  closed  polygon. 
In  any  neighborhood  of  Cx  there  will  be,  according  to  §  6,  on 
account  of  the  simple  connectivity,  an  elementary  curve  Cx 
lying  in  a  normal  subspace  of  two  dimensions 

(20)  Xi  =  gi(ui,  M2)  (i  =  1,  2,  •  •  -,  ra) 

entirely  interior  to  X.  If  the  continuation  curve  Cxy  is  not  closed, 
and  if  Cx  is  taken  sufficiently  near  to  Cx,  then  the  corresponding 
continuation  curve  Cxy  will  also  not  be  closed. 

The  normal  subspace  (20)  is  defined  over  a  simply  connected 
domain  U  of  points  (MI,  M2),  and  has  no  multiple  points.  To  every 
point  of  Cx  there  corresponds  therefore  a  single  pair  of  values 


(Cu)  u\  =  Mi(0,         M2  = 

and  the  functions  so  defined  are  continuous,  by  the  principal 
theorem  of  §  1,  since  at  every  point  some  pair  of  the  equations  (20) 
has  a  functional  determinant  for  MI,  M2  which  is  different  from 
zero.  The  curve  corresponding  to  the  curve  Cxy  in  the  space 
(M;  y)  may  be  denoted  by 

(Cuv)     ui  =  Mi(0,     M2  =  w2(0,     2/a  =  2/a(0,     (a  =  1,  2,  •  •  -,  n), 

and  its  initial  point,  corresponding  to  p',  by  pu'  (M';  y').  Every 
point  of  Cuv  is  an  ordinary  solution  of  the  equations 

(21)    <PO(MI,  ?/2;  yi,y<t,  •••,yn)=/«(0i,02,  •••,gm\  y\,y*,--,yn)  \\  0 

(a=  1,2,  >-,n). 


FUNDAMENTAL  EXISTENCE  THEOREMS.  37 

With  a  continuous  curve  C  joining  (u\,  Uz)  to  an  arbitrarily 
chosen  point  (u\,  u^)  of  U  there  is  always  associated  a  continu- 
ation curve  of  solutions  of  the  equations  (21),  having  the  initial 
point  puf  and  defined  throughout  the  whole  of  C,  since  any  such 
curve  defines  a  curve  in  the  a>space  interior  to  X  along  the  whole 
of  which  there  is  a  corresponding  continuation  curve  for  the 
equations  (19)  in  the  sheet  S.  Hence  there  is  a  unique  sheet  Su 
of  solutions  of  the  equations  (21)  whose  projection  in  the  UiUZ- 
space  is  U;  and  no  interior  point  of  U  is  a  point  where  the  sheet 
becomes  infinite  or  corresponds  to  a  boundary  point  of  the  sheet, 
since  the  same  is  true  of  5  with  respect  to  X,  The  preceding 
argument  can  therefore  be  applied  to  show  that  the  sheet  Su 
is  single-valued  over  the  region  U,  and  the  existence  of  the  curve 
Cuy  with  the  distinct  end  points  pu'  and  pu"  is  contradicted. 
Hence  Cxy  can  not  have  distinct  end  points  p'  and  p",  and  the 
theorem  last  stated  is  proved. 

§  8.    TRANSFORMATIONS  OF  n  VARIABLES  AND  A  MODIFICATION 
OF  A  THEOREM  OF  SCHOENFLIES 

It  is  interesting  to  deduce  by  means  of  the  preceding  theorems 
some  conclusions  concerning  a  system  of  equations  of  the  form 

(22)     fa(x;y)  =  xa-^a  (yb  y2,  •••,*/„)  =  0     (a  =  1,  2,  •  •  .,  n). 

The  functions  \J/  are  once  for  all  assumed  to  be  single-valued, 
continuous,  and  to  have  continuous  first  derivatives  in  a  con- 
tinuum Y  in  which  the  functional  determinant 


D  =  dtyi,  fa,  •  •  -,  <An)/d(?/i,  2/2,  •  •  •,  y») 

is  different  from  zero.  By  a  continuum  is  meant  a  set  of  points 
consisting  only  of  interior  points  any  two  of  which  can  be  con- 
nected by  a  continuous  curve  lying  entirely  within  the  set. 
The  boundary  points  of  Y  will  be  denoted  by  B,  and  X  will 
represent  the  set  of  points  in  the  z-space  which  corresponds  to  Y 
by  means  of  the  equations  (22). 


38  THE   PRINCETON  COLLOQUIUM. 

Any  sequence  \y(k)]  of  points  (yi(k),  y2(k),  •••,  yn(k)) 
(k  =  1,  2,  •  •  •)  in  Y,  which  approaches  infinity  or  has  a  point  of 
B  as  limit  point,  defines  a  corresponding  sequence  of  points 
\xw }  in  X.  The  set  of  points  of  condensation  for  such  sequences 
[x(k)]  will  be  denoted  by  A. 

The  totality  of  solutions  of  the  equations  (22)  corresponding  to 
points  of  the  continuum  Y  form  a  single  connected  sheet  S  whose 
only  boundary  points  have  projections  x  and  y  in  the  sets  A  and  B, 
respectively. 

For  suppose  that  (x'\  y')  is  a  first  solution  and  (x";  y"}  any 
other.  The  points  y'  and  y"  can  be  joined  by  a  continuous 
curve  interior  to  Y 

ya  =  y«(t)       («=i,  2,  ...,n;  t'^t^t"}, 

and  the  corresponding  curve 

*«  =  *a(0>        ya  =  ya(0, 

defined  by  equations  (22),  is  a  curve  interior  to  the  sheet  S  and 
joining  (x';  y')  to  (x"\  y"},  so  that  S  is  evidently  connected.  Any 
boundary  point  (a;j3)  of  <S  must  be  the  limit  of  a  sequence  of 
points  p(k)  for  which  the  projections  y  are  in  Y.  The  limit  /3 
of  the  sequence  y(k)  can  not  be  in  Y,  since  then  (a;  /3),  by  the 
theorem  of  §  1,  would  be  an  interior  point  of  S.  Hence  /3  must 
be  in  B  and  a  in  A. 

One  may  say  further  that  if  p(k)  is  a  sequence  of  points  (xw;yw) 
in  S  for  which  the  sequence  x(k)  approaches  infinity,  then  the 
only  finite  points  of  condensation  possible  for  the  sequence 
yW  are  jn  $.  The  statement  is  true  when  x  and  y  are  inter- 
changed, on  account  of  the  definition  above  of  the  set  A. 

If  the  points  of  the  set  A  are  distinct  from  those  of  the  image  X 
of  Y,  then  X  is  a  single  continuum  whose  only  boundary  points 
are  points  of  A. 

To  prove  this,  consider  an  arbitrarily  chosen  point  y'  of  Y. 
None  of  the  points  in  a  suitably  chosen  neighborhood  of  the 
corresponding  values  x'  are  points  of  A,  since  by  the  fundamental 


FUNDAMENTAL   EXISTENCE   THEOREMS.  39 

theorem  of  §  1  all  such  points  correspond  by  means  of  equations 

(22)  to  points  of  Y,  and  are  therefore  points  of  X.     Consider 
now  the  continuum  X  consisting  of  all  points  x  which  can  be 
joined  to  x'  by  continuous  curves  containing  no  points  of  A, 
a  continuum  to  which  the  neighborhood  of  x'  certainly  belongs, 
as  has  just  been  shown. 

All  the  points  of  X  are  in  the  continuum  X,  since  the  solutions 
of  equations  (22)  corresponding  to  points  of  Y  form  a  single 
connected  sheet  S.  The  curve  in  S  joining  (a/;  y')  with  any 
other  point  (x"\  y"}  of  the  sheet  has  therefore  a  projection  in 
the  a>space  joining  x'  with  x"  and  containing  no  points  of  the 
set  A. 

All  of  the  points  of  X  are  points  of  X.  For  any  set  of  values 
x  in  X  can  be  joined  to  x'  by  a  continuous  curve  Cx  lying  entirely 
in  X  and  containing  therefore  no  points  of  A.  By  the  second 
theorem  of  §  5  the  corresponding  continuation  curve  Cxy  must 
extend  along  the  entire  arc  Cx,  since  otherwise  the  values  of  y 
for  points  on  Cxy  would  approach  infinity  or  else  have  a  limit 
point  on  the  boundary  B  of  Y,  and  some  point  of  Cx  would  in 
that  case  necessarily  be  a  point  of  A.  It  follows  that  x,  like  x',  is 
the  image  of  some  point  y  in  Y. 

From  the  initial  theorem  of  the  last  section,  for  the  case  when 
there  are  more  than  two  variables,  it  follows  that 

If  A  is  distinct  from  X,  and  X  is  simply  connected  in  the  sense  of 
§6,  then  the  sheet  S  is  single-valued.  In  other  words  the  continuum  Y 
is  tranformed  in  a  one-to-one  way  into  a  continuum  X  by  means  of 
the  equations  (22),  and  the  functions 

(23)  ya  =  ya(xi,  3-2,  •••,  &„)       (a  =  1,2,  •  • -,  ri) 

so  defined  over  X  are  single-valued,  continuous,  and  have  continuous 
first  derivatives. 

The  character  of  the  functions  (23)  near  any  point  of  X  follows 
at  once  from  the  theorem  of  §  1. 

Let  it  be  supposed  that  the  set  of  points  A  divides  the  x-space 
into  exactly  tico  continua  X,  H  such  that  every  point  of  A  is  a  bound- 


40  THE   PRINCETON    COLLOQUIUM. 

ary  point  for  each  of  them,  and  suppose  furthermore  that  there  is  a 
particular  point  £  in  H  which  does  not  correspond  by  means  of  the 
equations  (22)  to  any  point  of  Y.  Then  the  image  X  of  Y  is 
distinct  from  A  and  coincides  with  X.  If  X  is  simply  connected 
the  other  conclusions  of  the  last  theorem  follow  at  once. 

In  the  first  place  it  can  be  shown  that  if  any  point  £'  of  H 
corresponds  to  a  point  of  Y  then  every  other  point  £"  of  E 
would  also  have  this  property.  For  £'  and  £"  can  be  joined  by 
a  continuous  curve 

xa  =  *.(/)         (a=l,  2,  .-.,  n;  t'  ^t  ^") 
entirely  interior  to  H.     The  corresponding  continuation  curve 


of  solutions  of  equations  (22)  must  be  defined  along  the  whole  of 
the  interval  t'  5^  t  ^  t",  since  otherwise  as  <  approached  any 
upper  bound  T  of  the  values  t  which  could  be  reached  by  con- 
tinuation, the  corresponding  points  y  of  the  curve  would  have 
to  approach  infinity  or  else  have  a  point  of  condensation  on  the 
boundary  of  Y.  But  this  is  impossible,  since  for  a  sequence  of 
points  x  corresponding  to  a  sequence  of  points  in  7  approaching 
infinity  or  a  boundary  point  of  Y,  the  only  limiting  points  possible 
are  at  infinity  or  else  in  the  set  A.  It  follows  at  once,  on  account 
of  the  hypothesis  of  the  theorem,  that  no  point  of  H  can  correspond 
to  a  point  of  Y,  and  neither  can  any  point  of  A,  since  in  any 
neighborhood  of  such  a  point  of  A  there  are  points  of  H  which 
in  that  case  would  also  correspond  to  values  y  in  Y.  The  image 
of  the  region  Y  in  the  z-space  is  a  single  continuum  whose  only 
boundary  points  are  points  of  A.  According  to  the  preceding 
argument  it  cannot  be  E  and  it  must  therefore  be  X. 

A  modification  of  a  theorem  of  Schoenflies  can  be  deduced 
readily  from  the  results  which  precede.  The  theorem  has  to  do 
with  a  pair  of  equations  of  the  form 

(24)  xi  -  fa(yi,  y2),        *2 


FUNDAMENTAL   EXISTENCE   THEOREMS.  41 

in  which  the  functions  \{/  are  single-valued,  continuous,  and  have 
continuous  derivatives  on  a  simply  closed  regular  curve  B  of  the 
y-plane  and  in  the  interior  7  of  B.  The  functional  determinant 
D  =  dtyi,  ^z)/d(yi,  2/2)  is  supposed  to  be  different  from  zero  in  Y. 
If  the  curve  A  in  the  x-plane  formed  by  transforming  the  simply 
closed  regular  curve  B  in  the  y-plane,  by  means  of  the  equations  (24), 
is  distinct  from  the  image  X  of  the  interior  Y  of  B,  then  X  is  a 
simply  connected  continuum  whose  only  boundary  points  are 
points  of  A,  and  the  correspondence  defined  between  X  and  Y  is 
one-to-one.  The  single  valued  functions 

(25)  2/1  =  y\(xi,  xz),        2/2  =  yz(xi,  x2), 

so  determined  in  the  region  X,  are  continuous  and  have  continuous 
first  derivatives.* 

From  the  preceding  theorems  of  this  section  it  follows  that 
the  complete  image  X  of  Y  is  a  single  finite  continuum  whose 
only  boundary  points  are  points  of  A.  It  remains  to  show  that 
X  is  simply  connected  and  that  the  correspondence  between  X 
and  Y  is  one-to-one. 

If  any  simply  closed  regular  curve  Cx  is  drawn  in  X,  its  interior 
must  consist  entirely  of  points  of  X.  Otherwise  there  would 
necessarily  be  a  boundary  point  of  X,  a  point  of  the  curve  A, 
interior  to  Cx,  and  there  would  also  be  points  of  A  exterior  to  Cx 
since  X  is  finite.  Hence  there  would  necessarily  be  a  point  of 
the  continuous  curve  A  on  Cx  itself,  which  contradicts  the  as- 
sumption that  A  and  X  are  distinct.  It  follows  at  once  from 
the  first  paragraphs  of  §  7  and  the  simple  connectivity  of  X  just 
proved,  that  only  one  point  y  in  Y  corresponds  to  a  given  x  in 
X,  and  by  the  theorem  of  §  1  it  may  be  seen  that  the  functions 

*  Schoenflies  assumed  only  the  continuity  of  the  functions  \l/i,  ^2,  adding, 
however,  that  the  correspondence  denned  between  the  regions  X  and  Y  of  the 
two  planes  is  to  be  one-to-one.  In  the  theorem  here  proved  \}>\  and  ^-2  are 
subjected  to  further  continuity  restrictions,  but  the  correspondence  is  proved 
to  be  unique.  See  Schoenflies,  "  Ueber  einen  Satz  der  Analysis  Situs," 
Gottinger  Nachrichten  (1899),  page  282.  The  theorem  was  later  proved  by 
Osgood  and  Bernstein  in  the  same  journal  (1900),  pages  94  and  98,  respectively. 


42  THE   PRINCETON   COLLOQUIUM. 

(25)  have  the  continuity  properties  described  in  the  theorem  in 
the  neighborhood  of  any  particular  point  x. 

Another  theorem,  slightly  different  in  form,  may  be  stated  as 
follows: 

If  the  images  of  the  points  of  the  simply  closed  regular  curve  B 
in  the  y-plane  all  lie  on  a  simply  closed  regular  curve  A  in  the  x-plane, 
then  the  equations  (24)  define  a  one-to-one  correspondence  between 
the  interior  X  of  A  and  the  interior  Y  of  B,  and  the  functions  (25) 
so  defined  have  the  same  continuity  properties  as  before. 

In  this  case  it  can  first  be  shown  that  the  image  x'  of  any  point 
y'  in  Y  must  be  distinct  from  A,  and  the  rest  of  the  proof  is  the 
same  as  before.  For,  if  x'  were  a  point  of  A,  every  point  of 
a  properly  chosen  neighborhood  of  x'  would  also  be  the  image  of 
a  point  of  Y,  since  at  (x';  y')  the  functional  determinant  of 
equations  (24)  does  not  vanish.  It  would  follow  then,  by  con- 
tinuation, that  every  point  exterior  to  the  curve  A  would  also 
be  the  image  of  a  point  of  Y,  which  is  impossible  since  thefunctions 
^  are  finite.  The  continuum  X  is  therefore  identical  with  the 
interior  of  A,  by  the  preceding  theorems,  and  the  correspondence 
between  X  and  Y  is  one-to-one. 

An  example  applying  some  of  the  theorems  of  §§  5,  8  is  given 
at  the  end  of  §  14. 


CHAPTER.  II 

SINGULAR  POINTS  OF  IMPLICIT  FUNCTIONS 

The  theorems  which  have  been  developed  in  the  preceding 
pages  of  these  lectures  have  to  do  with  the  behavior  of  implicit 
functions  at  ordinary  points,  or  in  regions  which  have  no  singular 
points  in  their  interiors.  For  singular  points  where  the  functional 
determinant  vanishes  the  theory  is  much  more  complicated,  and 
no  methods  which  can  be  comprehensively  applied  have  so  far 
been  developed.  There  are,  however,  many  special  cases  in 
widely  different  fields  which  have  been  studied  with  success, 
and  it  may  not  be  out  of  place  to  glance  at  a  few  of  them  before 
proceeding  to  the  further  theorems  with  which  these  pages  are 
primarily  concerned. 

Perhaps  the  most  complete  single  theory  which  has  been 
developed  is  that  which  has  to  do  with  the  singularities  of  an 
algebraic  function  y  of  x  determined  by  an  equation  of  the  form 

(1)  P(x,  y)  =  0, 

where  P  is  an  irreducible  polynomial  in  the  two  variables  x  and  y. 
Suppose  for  convenience  that  the  singular  point  to  be  considered 
is  at  the  origin,  and  that  the  polynomial  P(0,  y}  has  a  lowest 
term  of  degree  n  in  y.  Then  it  is  known  that  for  each  value  of 
z  in  a  sufficiently  small  neighborhood  of  x  —  0,  there  exist  exactly 
n  solutions  y  of  equation  (1)  in  the  neighborhood  of  y  =  0,  and 
the  values  of  these  solutions  are  given  by  k  cycles  of  the  form 

(2)  y=a^  +  a/z^+...     (j  =1,2,  •••,&), 

where  the  numbers  n,  p  are  positive  integers  satisfying  the 
relations 

[ij  <  m'  <  ju,"  <  •  •  • ,        p!  +  p2  +  •  •  •  +  pk  =  n. 

43 


44  THE   PRINCETON   COLLOQUIUM. 

The  series  is  one  member  of  the  cycle;  the  others  are  found  by 
replacing  xvp'  by  uvxllp> (v=  1,  2,  •••,  p,  —  1),  where  a;  is  a 
primitive  py-th  root  of  unity.  The  number  PJ  has  no  factor  in 
common  with  the  exponents  /i,-,  ju/,  •  •  • .  Otherwise  the  expansion 
would  be  in  terms  of  a  root  of  x  of  lower  order  than  PJ.  Thus 
there  are  in  all  n  series  in  fractional  powers  of  x  which  define  the 
roots  of  the  algebraic  equation  in  the  neighborhood  of  the  origin. 
The  coefficients  of  the  series  may  be  computed  by  means  of  the 
well-known  Newton  polygon,*  or  by  methods  due  to  Ham- 
burgerf  and  Brill. J  If  the  substitution  x  =  tK  is  made  in  the 
series  (2),  the  points  (x,  y)  which  it  defines  may  be  expressed 
in  the  parametric  representation 

x  =  f',         y  =  r>  [otj  +  a/P/-"'  +  >••}       (j  =  1,  2,  •  •  • ,  k). 

All  the  solutions  of  the  equation  (1)  in  the  neighborhood  of  the 
origin  evidently  belong  to  a  finite  number  of  such  branches. 

With  the  help  of  the  preparation  theorem  of  Weierstrass, 
which  is  to  be  studied  in  the  following  pages,  results  similar  to 
those  just  given  may  be  proved  for  the  solutions  of  an  equation 
F(x,  y)  =  0  in  the  vicinity  of  any  point  where  F  is  analytic. 

The  singularities  of  a  surface 

F(x,  y,  z)  =  0 

at  a  point  where  the  function  F  is  analytic  have  also  been  ex- 
tensively studied.  The  points  of  the  surface  in  the  neighbor- 
hood of  a  singular  point  are  determined  by  means  of  a  finite 
number  of  expansions  of  the  form 

x  =  P(u,  i>),        y  =  Q(u,  »), 

where  P  and  Q  are  analytic  in  the  parameters  u  and  «.§ 

*  See  Appell  and  Goursat,  Th6orie  des  Fonctions  algebriques,  pp.  184  ff. 

t  Weierstrass,  Werke,  vol.  4,  Kapitel  1. 

J  Munchener  Berichte,  vol.  21  (1891),  p.  207. 

§  See  Black,  "  The  parametric  representation  of  the  neighborhood  of  a 
singular  point  of  an  analytic  surface,"  Proceedings  of  the  American  Academy 
of  Arts  and  Sciences,  vol.  37  (1902),  p.  281. 


FUNDAMENTAL   EXISTENCE   THEOREMS.  45 

In  the  calculus  of  variations  the  construction  of  "  fields  of 
extremals  "  in  the  plane  requires  the  study  of  the  real  solutions 
of  a  system  of  equations  of  the  form 

(3)  *  =  <p(t,  a),        y  =  t(t,  a). 

The  extremals  are  the  curves  in  the  zy-plane  defined  by  these 
equations  for  different  values  of  o.  Suppose  that  the  parametric 
values 

(4)  to  ^  t  ^  ti,        a  =  a0 

define  an  arc  E  which  does  not  intersect  itself  and  which  consists 
entirely  of  points  where  the  functional  determinant 

(5)  A((- a)  • 

is  different  from  zero.  Then  to  any  point  (x,  y)  in  a  properly 
chosen  neighborhood  of  E  there  corresponds  but  one  solution 
(t,  a)  of  equations  (3),  in  the  neighborhood  of  the  values  (4);  and 
the  functions 

t  =  t(x,  y},        a  =  a(x,  y} 

so  defined  have  continuity  properties  similar  to  those  of  <p  and 
\f/  themselves.*  The  neighborhood  thus  simply  covered  by  the 
extremals  (3)  is  the  "field,"  and  is  perhaps  the  simplest  example 
of  the  notion  since  it  consists  only  of  non-singular  solutions  of  the 
equations  (3). 

When  it  is  desired  to  find  an  arc  C  which  minimizes  an  integral 
with  respect  to  variations  lying  entirely  on  one  side  of  C,  a  field  of 
a  different  sort  can  be  constructed.!  The  equations  of  the 

The  mathematical  literature  concerned  with  the  singularities  of  a  curve 
or  surface,  particularly  their  transformation  into  simpler  types,  is  very  large. 
The  reader  is  referred  to  Pascal,  Repertorium  der  hoheren  Mathematik,  2d 
edition,  vol.  2,  erste  Halfte,  pp.  291  ff ;  and  Encyclopadie  der  Mathematischen 
Wissenschaften,  II  B  2,  p.  119,  and  III  C  4,  pp.  365  ff. 

*  Bolza,  Vorlesungen  iiber  Variationsrechnung,  pp.  249  ff. 

t  Bliss,  "  Sufficient  conditions  for  a  minimum  with  respect  to  one-sided 
variations,"  Transactions  of  the  American  Mathematical  Society,  vol.-  5  (1904), 
p.  477:  Bolza,  "  Existence  proof  for  a  field  of  extremals  tangent  to  a  given 
curve,"  ibid.,  vol.  8  (1907),  p.  399. 


46 


THE   PRINCETON   COLLOQUIUM. 


extremals  (3)  can  be  taken  so  that  for  t  =  0  they  all  intersect  C 
and  are  tangent  to  it,  and  the  equations 

x  =  <f>(Q,  a),        y  =  iKO,  a) 

will  then  be  the  equations  of  C.  If  the  curvatures  of  the  two  arcs 
at  their  point  of  contact  are  always  different,  then  the  extremal 
arcs  E  simply  cover  a  portion  of  the  plane  N  on  one  side  of  C 
and  adjacent  to  it.  In  other  words,  the  equations  (3)  define  a 
one-to-one  correspondence  between  the  points  of  a  region  ad- 
joining the  axis  t  =  0  in  the  to-plane,  shown  in  the  accompanying 
figure,  and  a  certain  neighborhood  N  on  one  side  of  the  arc  C. 


FIG.  2. 

In  the  interior  of  the  region  N  the  functions  t(x,  y),  a(x,  y)  have 
continuity  properties  similar  to  those  of  <p  and  ^  themselves. 
It  is  easy  to  see  that  this  is  a  case  in  which  the  functional  de- 
terminant (5)  vanishes  along  the  boundary  t  =  0  of  the  region  to 
be  transformed,  since  the  curves  C  and  E  are  always  tangent. 
In  a  paper  published  since  these  lectures  were  given,  Dr.  E.  J. 
Miles*  has  considered  the  transformation  defined  by  the  equations 

*  "  The  absolute  minimum  of  a  definite  integral  in  a  special  field,"  Trans- 
actions of  the  American  Mathematical  Society,  vol.  13  (1912),  pp.  37  ff. 


FUNDAMENTAL   EXISTENCE   THEOREMS. 


47 


(3)  when  the  curve  C  to  which  the  extremals  E  are  tangent  has 
a  cusp,  a  situation  corresponding  to  still  another  problem  in  the 


FIG.  3. 

calculus  of  variations.  In  that  case  a  point  (t\,  ai)  and  a  curve 
F  through  it  are  transformed  into  a  point  (x\,  y\)  and  a  curve  C 
as  shown  in  the  figure.  One  portion  S  of  a  neighborhood  of 
(ti,  ai)  is  then  transformed  in  a  one-to-one  way  into  the  leaf  S, 
and  the  other  portion  S'  into  the  leaf  S'.  At  any  point  in  the 
interior  of  one  of  the  leaves,  the  variables  t  and  a  are  single- 
valued  functions  of  x,  y  having  continuity  properties  similar  to 
those  of  <p  and  \f/.  The  transformation  is  singular  along  the 
curve  F. 

The  three  examples  which  have  been  just  described  are  only  a 
few  of  the  many  proofs  for  the  existence  of  fields  involving  trans- 
formations with  singular  points  which  might  be  cited.*  Nearly 
all  of  these  have  to  do  with  singularities  of  transformations  of 
the  form 
(6)  x  =  <p(u,  v),  y  =  t(u,  v), 

*  Bliss,  "  The  construction  of  a  field  of  extremals  about  a  given  point," 
Bulletin  of  the  American  Mathematical  Society,  vol.  13  (1906),  p.  47;  Mason 
and  Bliss,  "  Fields  of  extremals  in  space,"  Transactions  of  the  American  Mathe- 
matical Society,  vol.  11  (1910),  p.  325;  Bill,  "  The  construction  of  a  space 
field  of  extremals,"  Bulletin  of  the  American  Mathematical  Society,  vol.  15 
(1908),  p.  374;  Sziics,  "Sur  Pextremale  qui  joint  deux  points  donnes,"  Mathe- 
matische  Annalen,  vol.  71  (1912),  p.  380.  The  method  used  by  Sztics  is  quite 
closely  that  of  Mason  and  Bliss  in  the  paper  mentioned  above. 


48  THE   PRINCETON  COLLOQUIUM. 

or 

x  =  <p(u,  v,  u,'),        y  =  $(u,  v,  w),        z=  x(u,  v,  w), 

which  have  been  studied  also  in  a  series  of  papers  of  more  recent 
date  presented  as  dissertations  for  the  degree  of  doctor  of 
philosophy  at  Harvard  University.*  The  methods  which  have 
been  used  in  the  different  cases  have  differed  widely,  and  it  does 
not  seem  possible  at  present  to  formulate  a  theory  which  includes 
them  all.  It  is  the  intention  of  the  writer,  however,  to  show  in 
the  following  pages  how  the  transformation  theorems  proved 
above  in  §  7  may  be  applied  to  throw  much  light  on  the  nature 
of  real  transformations  of  the  form  (6)  in  the  neighborhoods  of 
singular  points.  In  the  section  of  the  lectures  immediately 
following  this  introduction  a  simple  algebraic  proof  of  the 
preparation  theorem  of  Weierstrass  is  given,  not  depending 
upon  the  theory  of  functions  of  a  complex  variable.  A  general- 
ization of  it  is  given  in  a  later  section  which,  in  what  might  be 
called  the  general  case,  enables  one  to  describe  the  behavior  of 
the  solutions  of  a  system  of  equations  of  the  form 

/•(*i,  2-2,  •  •  -,  xm;  yi,  2/2,  •  •  •,  2/n)  =  0     (i  =  1,  2,  •  •  •,  n} 
in  the  neighborhood  of  a  point  where  the  functional  determinant 


2/2,  •••,  yn 

vanishes.     For  these  equations  the  variables  x  and  y  are  per- 
mitted to  have  complex  values.f 

*  Urner,  "  Certain  singularities  of  point  transformations  in  space  of  three 
dimensions,"  Transactions  of  the  American  Mathematical  Society,  vol.  13  (1912), 
p.  233;  Clements,  "  Implicit  functions  defined  by  equations  with  vanishing 
jacobian,"  to  appear  in  the  same  journal.  Dederick,  in  a  paper  entitled  "  The 
solutions  of  an  equation  in  two  real  variables  at  a  point  where  both  the  partial 
derivatives  vanish,"  Bulletin  of  the  American  Mathematical  Society,  vol.  16 
(1909),  p.  174,  has  discussed  the  singularities  of  a  curve  of  the  form  F(x,  y)  =  0 
with  the  help  of  a  sort  of  generalization  of  the  Weierstrass  preparation  theorem 
for  a  function  which  is  not  necessarily  analytic. 

t  The  proof  given  in  these  pages  for  the  last-mentioned  theorem  is  for  the 
case  of  two  variables  y.  For  n  variables  see  the  reference  in  the  last  footnote 
to  §  13. 


FUNDAMENTAL  EXISTENCE  THEOREMS.  49 

§  9.    THE  PREPARATION  THEOREM  OF  WEIERSTRASS 

The  theorem  which  is  to  be  proved  may  be  stated  in  the 
following  form: 

Letf(xi,  xz,  •  '  •  ,  xm,  y)  be  a  convergent  series  in  the  variables  x,  y, 
and  such  that  the  series  /(O,  0,  •  •  •  ,  0,  y}  begins  with  a  term  of  degree 
n.  Then  f  is  factorable  in  the  form 


where  a\,  a2,  •  •  •  ,  an  are  convergent  power  series  in  x\,  x%,  •  •  •  ,  xm 
which  vanish  for  Xi  =  x2  =  •  •  •  =  xm  =  0,  and  <p  is  a  power  series 
in  Xi,  Xz,  •  '  •  ,  xm,  y  which  has  a  constant  term  different  from  zero. 

In  the  Bulletin  de  la  Societe  Mathematique  de  France*  Goursat 
has  called  attention  to  the  fact  that  the  proof  which  Weierstrass 
gave  of  this  important  theorem,  as  well  as  the  later  proofs 
which  occur  in  the  literaturef,  make  use  of  the  notions  of  the 
function  theory,  while  the  theorem  itself  is  essentially  of  an 
algebraic  character.  In  the  paper  referred  to  he  has  given  an 
elegant  and  elementary  proof  of  the  theorem  which  is  in  outline 
as  follows: 

By  means  of  the  substitution 

yn  =  —  aiyn~l  —  a2yn~2  —  •  •  •  —  an 

the  series  /  can  be  reduced  to  a  polynomial  P  of  degree  n  —  1 
in  y,  whose  n  coefficients  are  convergent  series  in  a\,  a2,  •  •  •, 
an,  Xi,  x^  •  •  •  ,  xm.  By  the  usual  theorems  of  implicit  function 
theory  it  is  shown  that  the  n  equations  found  by  putting  these 
coefficients  equal  to  zero  have  unique  solutions  for  a\,  a2,  •  •  •  ,  an 
as  power  series  in  x\,  x2,  •  •  -  ,  xm,  which  vanish  with  xif  x2,  •  •  •,  xm. 
If  the  values  so  found  are  substituted  in  the  formula 

yn  =  —  aiyn~l  —  azyn~2  —  •  •  •  —  an  +  M 


*  "  Demonstration  elementaire  d'un  the'oreme  de  Weierstrass,"  vol.  36 
(1908),  p.  209. 

t  See,  for  example,  Picard,  Traite  d'Analyse,  vol.  2,   p.   243;  Goursat, 
Cours  d'Analyse,  vol.  2,  p.  284. 
5 


50  THE  PRINCETON   COLLOQUIUM. 

and  the  series/  again  reduced,  a  polynomial  PI  of  degree  n  —  1 
in  y  will  be  found  whose  coefficients  are  series  in  x\,  xz,  •  •  •  ,  fm,  M- 
On  account  of  the  way  in  which  the  functions  a\,  at,  •  •  •  ,  an 
were  determined,  this  polynomial  PI  has  a  factor  n,  and  hence 
/  has  a  factor  (yn  +  aiyn~l  +  •  •  •  +  a»). 

Since  the  paper  of  Goursat  appeared  two  further  proofs  of  the 
theorem  have  been  published,  one  by  the  writer*  and  the  other 
by  MacMillan.t  each  of  which  seems  even  more  direct  than  that 
of  Goursat.  In  the  proof  which  follows  use  is  made  of  the  very 
concise  and  elegant  method  of  MacMillan  for  determining  the 
coefficients,  wrhile  the  rest  of  the  proof  is  similar  to  that  of  the 
earlier  paper  of  the  writer  cited  above. 

The  theorem  may  be  stated  in  a  different  form  as  follows: 
Suppose  thatf(x\,  x-t,  •  •  •  ,  xm,  y)  is  a  series  with  literal  coefficients 
such  that  /(O,  0,  •  •  •  ,  0,  y]  begins  with  the  term  a0yn.     Then  there 
is  one  and  but  one  series  b(x\,  xz,  •  •  •,  xm,  y)  which  satisfies  formally 
the  relation 

(7)  bf  =  p, 

where  p  is  a  polynomial 

p  =  aGyn  +  a^yn~l  +  •  •  •  +  an 

whose  coefficients  Ok(xi,  x%,  •  •  •,  xm)(k  =  1,  2,  •  •  -,  n)  are  series 
vanishing  with  the  x's. 

Each  of  the  coefficients  in  b  and  the  a's  is  a  rational  function  of  a 
finite  number  of  the  coefficients  of  f  with  denominator  a  power  of 
OQ,  and  the  constant  term  in  b  is  unity. 

If  the  coefficients  in  f  are  chosen  numerically  so  that  f  converges 
and  a0  4=  0,  then  the  series  b  and  a*  (k  =  1  ,  2,  •  •  •  ,  n)  also  converge. 

The  functions  /,  b,  p  may  be  written  in  the  forms 


/  =  a0yn  -  yn+lf0  -  /i  -  /2  - 
(8)  b  =  b0+  &!  +  62+  .-., 

p  =  a0yn  —  pi  —  pz—  ••-, 

*  Bulletin  of  the  American  Mathematical  Society,  vol.  16  (1910),  p.  356. 
t  Ibid.,  vol.  17  (1910),  p.  116. 


FUNDAMENTAL  EXISTENCE  THEOREMS.  51 

where  fk,  bk,  pk  are  homogeneous  expressions  of  degree  k  in 
%i,  %2>  '  *  •  >  %m  with  coefficients  which  are  series  in  y.  It  is  desired 
to  determine  b  so  that  the  identity  (7)  holds,  and  so  that  the 
expressions  pk  have  coefficients  which  contain  y  only  to  the  degree 
n-  1. 

By  substituting  the  expressions  (8)  in  the  identity  (7)  and 
equating  terms  of  the  same  degree  in  the  z's,  it  follows  that 

bo(a0  —  yfQ}yn  =  a0yn, 

bi(a0  —  yf0)yn  =  60/i  —  Pi, 

bz(a0  —  yfo)yn  =  60/2  +  &i/i  —  Pz, 

' ) 
bk(a0  —  yfo)yn  =  b0fk  +  bifk-i  +  •  •  •  +  &t-2/2  +  bk-ifi  —  Pk, 

These  equations  are  to  be  identities  in  x  and  y.  The  first  one 
determines  60  uniquely  with  constant  term  unity,  and  further- 
more so  that  each  coefficient  is  a  quotient,  in  fact  a  polynomial 
with  positive  integral  coefficients  in  a  finite  number  of  the  coef- 
ficients of  /,  divided  by  a  power  of  a0.  In  the  second  equation 
pi  must  be  chosen  equal  to  the  terms  of  60/i  which  contain  y 
to  the  degree  n  —  1  or  less,  after  which  61  is  uniquely  determined. 
Similarly  in  the  fcth  equation  pk  must  first  be  chosen  to  cancel 
the  terms  on  the  right  of  degree  n  —  1  or  less  in  y,  and  then  bk 
is  unique. 

It  only  remains  to  show  that  the  series  6  and  a&  are  convergent 
in  any  numerical  case  for  which  /  converges.  There  is  no  loss 
of  generality  in  assuming  that  the  series  /  converges  in  the  domain 

\Xi\  ^1,        \y    ^1         (i  =  1,  2,  •••,  m), 

since  this  can  always  be  effected  by  a  substitution  of  the  form 

Xi  =  piXi,       y  =  try'        (i  =  1,  2,  •  •  -,  TO). 

Suppose  then  that  K  is  a  number  greater  than  the  absolute 
value  of  any  term  in  the  series /(I,  1,  •  •  •,  1,  1),  that  is,  greater 


52  THE   PRINCETON   COLLOQUIUM. 

than  the  absolute  value  of  any  coefficient  in  /.     If  A0  is  the 
absolute  value  of  ao,  the  series 

n  _  Ryn+l  _    K 

where 


dominates  /  in  the  sense  that  every  coefficient  except  the  first 
has  a  numerical  value  equal  to  or  greater  than  K;  and  the  series 
B  satisfying  the  relation 

BF  =  A0yn  +  A,yn~l  +  •  •  •  +  An 

analogous  to  (7)  has  coefficients  numerically  greater  than  the 
absolute  values  of  those  of  b.  Hence  if  B  converges  the  same 
will  be  true  of  6. 

But  it  is  easy  to  show  that  the  series  B  converges.  It  will 
certainly  do  so  if  convergent  series  Ak,  C,  D  can  be  found  satisfy- 
ing the  relation 

Aoyn(l-y)-Kyn+l-KX=(AQyn+Aly*-l+  •  •  •  +An)(Cy  +  D), 
because  then  B  would  have  the  value 


Cy+D' 

On  comparing  the  coefficients  of  the  two  highest  terms  in  y 
in  the  next  to  last  equation,  and  for  convenience  denoting  by  a 
the  constant  value 


~AT> 
it  is  found  that 

C  =  aA0,        D  =  1  -aAi. 
By  comparing  the  other  powers  of  y  and  substituting  these  values, 


FUNDAMENTAL  EXISTENCE  THEOREMS.  53 


we  have 

AI  +  a 

AI  + 


An-i  +  aAQAn  =  aAiAn-i, 

An  =  aA^An  -  KX. 

But  these  equations  have  linear  terms  in  AI,  A%,  •  •  •,  An  with 
functional  determinant  different  from  zero,  and  hence  have 
solutions,  by  the  theorems  of  §  2,  which  are  convergent  series  in 
Xi,  x2,  •  •  • ,  xm  and  have  no  constant  terms. 

It  is  evident,  in  any  numerical  case  for  which  /  is  convergent, 
that  a  neighborhood  of  the  origin  may  be  chosen  in  which  the 
series  b  is  everywhere  different  from  zero.  In  such  a  neighbor- 
hood all  of  the  values  (xi,  x2,  •  •  •,  xm,  y]  which  make/  vanish  are 
roots  of  the  equation  p  =  0,  and  vice  versa. 

If/Gri,  0,  •  •  •,  0,  y)  has  its  terms  of  lowest  degree  homogeneous 
and  of  degree  n,  then  the  polynomial  p(xif  0,  •  •  •,  0,  y]  has  the 
same  initial  terms,  since  the  first  coefficient  of  the  factor  series 
6  is  unity. 

§  10.    THE  ZEROS  OF  <p(u,  v),  \f/(u,  v),  OR  THEIR  FUNCTIONAL 
DETERMINANT 

Consider  a  function  <p(u,  v)  whose  values  in  the  neighborhood 
of  the  origin  in  the  m'-plane  are  given  by  a  convergent  series  in 
u  and  v  which  vanishes  for  u  =  v  =  0.  If  the  series  contains  a 
factor  u  in  every  term  it  may  be  written  in  the  form 

(9)  <p(u,  v)  =  auk$(u,  v), 

where  a  is  a  constant  different  from  zero  and  $(u,  v)  is  a  con- 
vergent series  for  which  <l>(0,  »)  has  a  first  term  of  the  form  im 
with  coefficient  unity.  According  to  the  results  of  the  preceding 
section,  all  of  the  roots  of  $(u,  v)  in  a  neighborhood  of  the  origin 
will  be  roots  of  a  certain  polynomial 

(10) 


54  THE   PRINCETON  COLLOQUIUM. 

where  the  coefficients  a*  are  series  in  u  having  no  constant  terms. 
The  polynomial  P  may  be  equal  to  the  product  of  two  poly- 
nomials of  similar  form, 

boVk  -f-  61^*— l  -{-  •  •  •  -}-  bk, 

C0Tm~k  +  Citfn~k~l  +    •  •  •  + 


where  the  coefficients  b  and  c  are  convergent  series  in  u.  In 
that  case  the  product  boCo  must  be  identically  unity,  and  by 
dividing  the  first  polynomial  by  60  and  multiplying  the  second 
by  the  same  series,  the  two  factors  will  have  the  form 

«*  +  fciV-i  +  -  -  -  +  bk', 

----  h  cm.k', 


The  coefficients  b'  and  c'  are  now  series  in  u  without  constant 
terms.  Otherwise  the  product  P  would  have  a  term  of  lower 
degree  than  tf",  with  a  coefficient  series  whose  constant  term 
would  be  different  from  zero. 

It  is  readily  seen  from  this  that  the  polynomial  P  is  either 
irreducible  in  the  sense  that  it  can  not  be  decomposed  into  a 
product  of  polynomials  of  the  same  sort,  or  else  it  is  the  product 
of  a  number  of  irreducible  polynomials  of  lower  degree. 

Suppose  that  Q(u,  r)  is  a  polynomial  of  the  form  (10)  which  is 
irreducible  in  the  sense  just  described.  Then  its  discriminant 
with  respect  to  r  is  a  series  in  u  which  does  not  vanish  identically, 
since  otherwise  Q  and  Qv  would  necessarily  have  a  common  factor 
of  the  form  (10),  and  Q  would  not  be  irreducible.  There  is  a 
neighborhood  0  <  u  ^  u\  in  which  the  discriminant  is  every- 
where different  from  zero,  and  for  any  value  u  satisfying  these 
inequalities  the  values  of  v  making  Q  =  0  are  all  distinct. 
According  to  the  results  which  have  been  stated  above  in  the 
introduction  to  this  chapter  of  the  lectures,  the  values  of  v  which 
make  Q  vanish  for  different  values  of  u  will  be  defined  by  m 
series  of  the  form 

(11)  r  =  aw(t/p  +  aV/p+  •••; 


FUNDAMENTAL  EXISTENCE  THEOREMS.  55 

and  these  series  must  all  be  distinct,  since  for  sufficiently  small 
values  u  4=  0,  as  has  been  seen,  the  roots  of  Q  are  all  distinct.* 
It  is  evident  then  that  all  the  roots  of  <f>(u,  v)  in  the  neighbor- 
hood of  the  origin,  including  those  which  correspond  to  the  fac- 
tor uk  in  equation  (9),  are  given  by  a  finite  number  of  elements 
of  the  form 

u  =  atp,        v=  fa"  +  &'r'+  •••, 

where  a  and  b  do  not  vanish  simultaneously,  and  p,  p,  \i! , 
are  positive  integers  having  no  common  factor. 
The  product  of  factors  of  the  form 

(12)  {v  -  cmu/p  -  a'u»'lp  -  •  •  •}, 

corresponding  to  the  elements  of  a  cycle,  is  a  polynomial  Q\(u,  v) 
of  the  form  (10).  For  the  product  Qi  is  a  series  in  ullp  and  v 
which  is  unchanged  when  ullp  is  replaced  by  <t)"ullp,  and  Qi  must 
therefore  contain  only  powers  of  ullp  whose  exponents  are  multi- 
ples of  p,  that  is,  positive  integral  powers  of  u. 

On  the  other  hand  an  irreducible  polynomial  Q  possesses  only 
a  single  cycle  of  elements  of  the  form  (12).  Each  element  of  a 
cycle  belonging  to  Q  gives  rise,  in  fact,  to  a  factor  Qi  of  Q  of 
the  form  (10).  The  number  of  elements  in  the  cycle  could  not 
be  greater  than  the  degree  of  Q,  and  neither  could  it  be  less, 
since  according  to  the  argument  of  the  paragraph  just  preceding, 
Q  would  then  be  divisible  by  a  factor  of  the  same  form  corre- 
sponding to  the  product  of  the  factors  (12)  belonging  to  the  cycle. 

By  combining  these  two  results,  it  follows  that  the  product  of 
the  factors  of  the  form  (12)  corresponding  to  the  elements  of  a  single 
cycle  is  an  irreducible  polynomial  of  the  form  (10),  and  conversely 
the  elements  of  an  irreducible  polynomial  of  the  form  (10)  form  a 
single  cycle. 

The  Weierstrassian  polynomial  P  of  any  function  <p  is  a  product 
of  irreducible  factors  of  the  same  form,  some  perhaps  repeated, 

*  The  method  of  proof  for  this  statement  in  the  case  of  a  polynomial  P 
is  precisely  that  of  the  theory  of  algebraic  functions.  See  the  reference  above 
(page  44)  to  Appell  and  Goursat. 


56  THE   PRINCETON   COLLOQUIUM. 

to  each  of  which  there  corresponds  a  cycle  of  elements.  By  the 
order  of  an  element  of  <f>  is  meant  the  number  of  times  its  factor 
(12)  is  repeated  in  the  product  ukP.  The  order  is  evidently 
equal  to  the  multiplicity  in  ukP  of  the  irreducible  factor  to  which 
the  element  belongs.  If  <p  possesses  one  element  of  a  cycle  it 
must  possess  the  whole  cycle.  For  the  polynomial  P  belonging 
to  <p  has  then  a  common  factor  with  the  irreducible  polynomial 
Q  of  the  cycle,  and  so  must  be  divisible  by  Q. 

Suppose  now  that  <p(u,  »)  and  \f/(u,  v)  are  two  functions  of  the 
form  described  above,  and  that  the  functional  determinant 


(13)  D(u,  v)  = 


does  not  vanish  identically. 

//  <p  and  \l/  have  an  element  in  common,  then  they  have  in  common 
the  irreducible  polynomial  Q  of  the  form  (10)  to  which  the  element 
belongs,  and  Q  is  also  factor  of  D. 

The  first  part  of  this  statement  follows  from  the  preceding 
paragraphs,  so  that  <p  and  ^  may  be  supposed  to  have  the  forms 

<p=  QA,  t  =  QB. 

When  these  expressions  are  substituted  in  the  functional  de- 
terminant (13)  the  presence  of  the  factor  Q  is  at  once  evident. 

A  similar  argument  shows  that  if  <p  has  an  element  with  cor- 
responding factor  Q  of  multiplicity  k,  and  \f/  has  the  same  element 
and  factor  with  multiplicity  I,  then  D  contains  the  element  and  its 
factor  with  multiplicity  k  -\-  I  —  1  at  least. 

There  is  a  sort  of  converse  to  these  statements  to  the  effect 
that  when  <p  and  D  hate  an  element  and  its  factor  Q  in  common,  then 
the  element  and  Q  are  either  multiple  in  <p  or  else  are  common  to  <p 
and  \f/. 

To  prove  this  let 

v  =  QA,      D  =  qc, 


FUNDAMENTAL   EXISTENCE  THEOREMS.  57 

and  suppose  Q  not  a  multiple  factor  of  <p.     Then 

=  QC; 
and  it  follows  readily  that  the  determinant 

(14) 

has  the  factor  Q,  since  A  can  not  have  any  element  in  common 
with  Q.  Otherwise  it  would  contain  the  whole  irreducible  factor 

Since  Q  is  irreducible,  its  discriminant,  a  series  in  u,  can  not 
vanish  identically,  and  there  is  an  interval  0  <  u  ^  u\  in  which 
it  is  different  from  zero.  For  any  value  of  u  satisfying  these 
inequalities  the  polynomials  Q  and  Q0  have  no  common  root.  If 

(15)  u  =  atP,         x  =  at*  +  a't*'  +  •  •  • 

is  the  parametric  form  of  one  of  the  elements  of  Q,  then  Q(u,  v) 
vanishes  identically  in  t  when  these  expressions  are  substituted, 
and  Qv(u,  v)  is  not  identically  zero  in  t  along  the  element.  Hence 
there  is  an  interval  0  <  t  ^  t\  in  which  Qv  is  different  from  zero. 
Since  the  determinant  (14)  has  the  factor  Q  and  therefore  vanishes 
identically  along  the  curve  (15),  it  follows  that 

du 

is  an  identity  in  t.  Evidently  if/(u,  v)  must  be  constant  along 
the  element,  and  its  value  is  everywhere  zero  since  it  vanishes 
for  t  =  0.  Hence  \f/  has  the  element  (15)  in  common  with  Q, 
and  must  have  Q  itself  as  a  factor  since  Q  is  irreducible. 

The  real  points  (u,  v)  where  one  or  another  of  the  functions 
<p,  \l/,  D  vanishes  play  an  important  role  in  the  investigation 
which  follows.  In  the  discussion  of  them  which  follows  it  will 
always  be  understood  that  when  u  is  real  and  positive  the  symbol 
w1/p  stands  for  the  real  and  positive  pth  root  of  u. 


58  THE   PRINCETON  COLLOQUIUM. 

If  the  function  <p  has  no  factor  u,  and  if  each  of  its  elements 
when  written  in  the  form 

(16)  v  =  u*1"  {a  +  a'u^'-^p+  •••} 

has  at  least  one  imaginary  coefficient,  then  in  a  neighborhood  of 
the  origin  no  real  point  (u,  0)  with  u  >  0  satisfies  the  equation 
<p(u,  v)  =  0. 

To  show  this,  suppose  for  the  moment  that  a.  is  imaginary. 
Then  for  sufficiently  small  positive  values  of  u  the  absolute  value 
of  a'u(li'~^lp  +  •  •  •  will  be  less  than  the  absolute  value  of  the 
imaginary  part  of  a,  and  the  parenthesis  in  the  expression  (16) 
will  also  be  imaginary.  A  similar  argument  would  show  v  to  be 
complex  if  one  of  the  higher  coefficients  were  the  first  not  real. 

On  the  other  hand,  if  the  coefficients  in  the  expression  are  all 
real,  then  for  positive  values  of  u  the  values  of  v  are  real,  and  the 
points  (u,  v)  so  defined  lie  on  a  real  arc  of  the  form 

u  =  tp,      v  =  «r  +  a'r'  +  "•    (o  <;  t  ^  *i). 

If  the  elements  of  <p  are  written  in  the  form 

(17)  v  =  a?(-  uYlp  +  «V(-  w)v/p  +  •  •  •, 

where  e  is  a  fixed  pih  root  of  —  1,  then  an  argument  similar  to 
that  just  given  shows  that  <p  =  0  is  satisfied  by  no  real  points 
in  the  neighborhood  of  the  origin  with  negative  values  of  u, 
unless  at  least  one  of  the  expressions  (17)  in  (—  u)llp  has  all  of 
its  coefficients  real.  On  the  other  hand  any  such  element  with 
real  coefficients  defines  points  (u,  v)  on  a  real  arc 

u  =  -tp,        v  =  £r  +  0YM  +  •  •  •      (0  ^  t  ^  <i). 

By  combining  these  results  it  follows  that  all  of  the  real  points,  in  a 
neighborhood  of  the  origin,  which  satisfy  <f>(u,  «)  =  0,  are  the 
points  of  a  finite  number  of  distinct  elements  of  the  form 

(18)  u  =  at',        r  =  6r  +  6Y*1  +  •  •  •      (0  ^  *  ^  <0 


FUNDAMENTAL   EXISTENCE   THEOREMS.  59 

whose  coefficients  are  real  and  such  that  a  and  b  are  not  both  zero. 
It  may  be  of  interest  to  note  in  passing  that  if  an  element  of  <p 
of  the  form  (16)  has  real  coefficients,  then  the  irreducible  poly- 
nomial Q  which  belongs  to  that  element  is  real.  For  Q  is  the 
product  of 

{v  —  au»lp  —ct'ull'lp  —  -••} 

and  the  other  factors  which  arise  from  it  by  replacing  ullp  by 
<a"ullp(v  =  0,  1,  2,  •  •  •,  p  —  1).  The  coefficients  of  the  product 
are  therefore  rational  integral  functions  with  real  coefficients 
in  the  a's  and  the  pth  roots  of  unity,  and  symmetric  in  the  latter. 
But  symmetric  functions  of  the  pth  roots  of  unity  are  real.  A 
similar  remark  holds  true  for  the  real  elements  of  the  form  (17). 

Two  real  elements  of  the  form  (18)  are  said  to  be  distinct  if  there 
is  an  interval  0  <  t  ^  t\  on  which  the  points  (u,  v)  which  they  define 
are  all  distinct.  Any  two  elements  are  either  distinct  or  else  coin- 
cident throughout. 

Let  the  two  elements  have  the  equations 

u  =  atp,      v  =  6r  +  6'r'  +  ~-    (0  5;  t  ^  ti), 

u  =  &*,        v=  dt"  +  d'V'  +  •  •  -     (0  ^  t  ^  <2). 

If  a  =  c  =  0  then  the  elements  are  distinct  unless  b  and  d  have 
the  same  sign,  in  which  case  each  defines  the  same  half  ray  from 
the  origin  along  the  r-axis.  If  a  =  0,  c  =J=  0  the  elements  are 
distinct.  If  a  and  c  are  both  different  from  zero  then  the  elements 
are  distinct  unless  the  expressions 


"IP  /  ti  \  "'IP 


•-'(?)*+'(:) 


are  identical  in  fractional  powers  of  u,  in  which  case  the  two 
elements  coincide. 

It  can  readily  be  seen  that  if  two  functions  <p  and  ^  have  a 
real  element  in  common  then  they  must  each  contain  the  irreduci- 
ble real  factor  which  belongs  to  the  element. 


60  THE    PRINCETON    COLLOQUIUM. 

§11.    SINGULAR  POINTS  OF  A  REAL  TRANSFORMATION  OF  Two 

VARIABLES 

In  this  section  it  is  proposed  to  study  the  singular  points  of  a 
transformation 
(19)  x  =  <p(u,  v),        y  =  $(u,  v) 

for  which  <p  and  \f/  are  convergent  series  in  u,  v  with  real  coef- 
ficients. It  is  presupposed  that  the  functional  determinant  D 
of  <p  and  \(/  does  not  vanish  identically,  and  that  the  real  elements 
of  <p  and  ^  described  in  §  10  are  all  distinct.  There  is  an  interval 
0  ^  t  ^  ti  for  which  the  elements  of  <p,  \l/,  and  D  which  are 
distinct  have  only  the  point  (u,  v)  =  (0,  0)  in  common.  Some 
of  these  elements  may  belong  to  both  <p  and  D,  or  to  ^  and  D, 
but  none  are  common  to  <p  and  \l/.  By  further  restricting  the 
interval  if  necessary,  it  can  be  effected  that  the  radius 


constantly  increases  on  each  element  as  t  increases  from  0  to  ti. 
For  p  is  a  series  in  t  which  does  not  vanish  identically,  and  its 
derivative  has  the  same  character.  An  interval  0  <  t  ^  t\  can 
therefore  always  be  selected  on  which  both  p  and  dpjdt  remain 
greater  than  zero. 

It  follows  immediately  that  a  constant  p\  can  be  selected  so 
that  any  circle  about  the  origin  of  radius  pi  or  less  is  intersected 
once  and  but  once  by  each  of  the  elements  in  question.  The 
real  elements  of  <p,  \f/,  and  D  may  therefore  be  represented  as 
shown  in  Fig.  4. 

//  the  value  of  p\  is  properly  restricted  then  any  one  of  the  regions 
S  shown  in  the  figure  is  transformed  in  a  one-to-one  way  by  the 
equations  (19)  into  a  region  S  adjoining  the  origin  and  lying 
entirely  in  one  quadrant  of  the  xy-plane.  The  single-valued  inverse 
functions 
(20)  u  =  f(x,  y),  v=  g(x,  y} 

so  defined  are  continuous  oner  all  of  2  and  analytic  in  its  interior. 


FUNDAMENTAL   EXISTENCE   THEOREMS. 


61 


To  prove 
by  the  equations 


this  consider  the  functions  r(u,  v)  and  co(w,  v)  defined 


VO         I 
<P  + 


TT  •  TT 

cos  co  =  ~  ,    sin  co  =  - 

T  T 


If  the  radius  pi  is  properly  restricted,  then  r  and  co  (modulus  2ir) 
are  well  defined  at  every  point  of  the  circle  with  the  exception 
of  the  origin,  since  <p  and  \l/  have  no  real  roots  in  common  aside 
from  (u,  v)  =  (0,  0). 

The  value  of  r  increases  monotonically  along  any  analytic  curve 

for  which  u  and  v  are  not  identically  zero,  as  may  be  seen  by 
reasoning  similar  to  that  applied  above  for  p,  after  noting  that  the 
series  for  <p  and  ^  can  not  vanish  identically  in  t.  In  particular 


FIG.  4. 


FIG.  5. 


if  pi  is  sufficiently  small,  then  r  has  this  property  along  the  bound- 
aries OEi  and  OE%  of  S,  and  along  an  auxiliary  arc  OE  chosen 
arbitrarily  for  purposes  of  proof  between  the  two  elements  OE\ 
and  OEZ. 

Suppose  now  that  k\  is  the  minimum  of  r  along  the  arc  £"1^2, 
and  select  arbitrarily  a  value  k  between  0  and  ki.     The  first  of 


62  THE  PRINCETON  COLLOQUIUM. 

the  equations 

(21)  r(u,  v)  =  k,        <o(w,  v)  =  z 

is  satisfied  at  a  unique  point  P(u0,  r0)  on  the  arc  OE,  and  the 
corresponding  value  of  z  may  be  denoted  by  z0.  The  functional 
determinant  of  r  and  co  has  the  value 

d(r,  «)  =  Db±9) 
d(u,  v)  r 

and  does  not  vanish  anywhere  in  the  interior  of  S. 

The  domain  in  which  the  equations  (21)  are  to  be  studied  is 
that  consisting  of  points  (u,  v,  z)  for  which  («,  »)  is  in  S,  and  z 
has  any  real  value.  According  to  the  first  theorem  of  §  5  and 
the  results  of  §  2  the  equations  (21)  define  two  analytic  functions 

(22)  u  =  M(Z),        v  =  v(z) 

which  take  the  initial  values  u0,  VQ  when  z  =  z0,  and  which  may 
be  continued  over  an  interval  z0  ^  2  <  f  ",  as  described  in  §  5. 
If  f"  is  the  value  defining  the  largest  such  interval,  the  points 
(u(z),  0(2))  corresponding  to  interior  points  of  the  interval  will 
all  be  interior  to  S,  while  as  z  approaches  f"  the  only  limit 
points  of  the  values  (u(z),  0(2))  must  lie  on  the  boundary  of  S. 
Otherwise  the  curve  (22)  could  be  continued  beyond  the  value  f ". 

The  length  of  the  interval  20  ^  2  <  f "  is  certainly  less  than 
ir/2,  since  in  the  region  <S  neither  sinco  nor  cosw  can  vanish. 
The  curve  (22)  can  not  intersect  itself,  since  the  same  values  of 
(u,  v)  must  define  the  same  z  by  means  of  the  second  of  equations 
(21). 

As  2  approaches  f  ",  the  point  (u(z),  0(2))  approaches  a  unique 
limiting  point  on  OEi  or  0J52.  This  follows  because  at  any 
limit  point  the  value  of  r(u,  v)  would  have  to  be  k,  and  this  can 
happen  at  one  point  PI  only  of  PE^,  and  at  one  point  PI  only 
of  OE-L.  The  curve  could  not  have  both  PI  and  P^  as  limit 
points  as  2  approaches  f  ",  since  then  it  would  necessarily  cross 
the  arc  OE  at  the  only  point  P  where  r(u,  v)  =  k,  and  so  would 
intersect  itself. 


FUNDAMENTAL  EXISTENCE  THEOREMS.  63 

A  similar  argument  shows  that  the  equations  (21)  define  an 
arc  without  double  point  over  an  interval  f '  <  z  ^  z0,  joining 
P  with  that  one  of  the  points  PI,  P2  which  was  not  the  end  of 
the  first  arc.  For  convenience  it  may  be  assumed  that  f  is 
the  value  belonging  to  PI,  and  f"  that  for  P2.  The  preceding 
inequalities  for  z  would  only  be  reversed  if  the  opposite  were  the 
case. 

There  are  no  other  points  in  the  region  S  at  which  r(u,  n}  =  k 
besides  those  of  the  arc  PiP2  which  has  just  been  defined.  If 
there  were  one  not  on  PiP2,  it  would  give  rise  to  a  second  curve 
of  the  same  sort  joining  PiP2.  But  this  new  curve  would 
necessarily  intersect  the  arc  OE  at  P,  and  hence  must  coincide 
with  the  original  arc  PiP2  throughout. 

For  any  value  k'  <  k  there  is  a  curve  similar  to  PiP2  on  which 
all  of  the  points  (u,  v)  making  r(u,  i>)  =  k'  lie. 

By  means  of  these  results  it  can  now  be  shown  that  any  two 
distinct  points  of  the  region  OP\Pz  are  transformed  into  two 
distinct  points  of  the  xy-plane.  For  if  (u',  «')  and  (u",  v") 
defined  the  same  point  (xf,  y'}  they  would  both  give  r  =  ^lx2-\-y2 
the  same  value  k',  and  hence  must  lie  on  the  same  curve  PiP2. 
But  in  that  case  the  values  of  w  corresponding  to  the  two  points 
would  necessarily  be  different,  as  has  been  seen  above,  and  hence 
(xr,  y')  and  (x",  y")  could  not  be  the  same. 

From  the  final  theorem  of  §  8  it  follows  at  once  that  the  theorem 
last  stated  above  is  true,  provided  that  the  circle  of  radius  p\ 
is  altered  so  that  the  arc  of  it  which  lies  between  the  branches 
OEi  and  OE%  lies  also  within  the  region  0PiP2.  The  region 
into  which  S  is  transformed  must  lie  entirely  in  one  quadrant 
of  the  xy-plane,  since  the  values  of  co  which  correspond  to  points 
of  S  are  all  in  one  quadrant.  In  the  interior  of  the  image  of  S 
the  inverse  functions  (20)  are  analytic,  since  at  interior  points  of 
S  the  determinant  D  is  different  from  zero. 

Some  conclusions  with  regard  to  the  distribution  of  the 
elements  of  <p,  \f/,  and  D  can  be  readily  derived  from  the  dis- 
cussion just  preceding.  For  example,  no  region  S  can  be  bounded 


64  THE   PRINCETON   COLLOQUIUM. 

by  two  elements  of  tp.  If  it  were  not  so,  then  in  a  region  bounded 
by  two  elements  of  <p  the  value  of  o>  on  the  branch  OE\  would 
be  everywhere  ir/2,  or  else  everywhere  —  7r/2,  and  the  same  is 
true  for  OE%.  But  this  is  impossible  since  along  the  arc  P\Pi 
the  value  of  w  varies  monotonically  through  an  interval  less 
than  7T/2.  A  similar  remark  holds  for  the  elements  of  \l/.  Hence 
it  follows  easily  that 

Between  any  elements  of  D  the  elements  of  <p  and  \f/,  if  there  are 
any,  must  separate  each  other. 

If  the  determinant  D  has  opposite  signs  in  two  adjoining 
regions  S  and  Sf  of  the  circle  of  radius  p\  in  the  wt'-plane,  shown 
in  Fig.  5,  their  transforms  in  the  a-^-plane  will  be  folded  over 
the  image  of  the  curve  OE2  and  will  overlap.  In  order  to  prove 
this,  let  it  first  be  remembered  that  along  the  element  OE2 

dr          du          dv 


so  that  ru  and  rv  can  not  vanish  at  any  point  P2  different  from 
the  origin.  Neither  can  they  vanish  at  an  interior  point  of 
one  of  the  regions  S,  since  at  a  point  where 


ru  = 


u          A  <P<Pv  v          n 

-  =  0,         rv  =  -    ——    -  =  (J, 


the  determinant  D  would  necessarily  have  the  value  zero,  and 
this  does  not  occur  in  the  interior  of  S.     The  equations 

du          dv  du          dv 

fu  T~  H~  fv  -j  =  0,        cou  -j-  +  cor  -j-  =  1 
dz          dz  dz          dz 

are  satisfied  everywhere  between  PI  and  P2  on  the  arc  (22). 
Hence 

du  _        r  dv  _   r 

Tz=    ~l)rv'  Iz  =  DTu' 

As  z  approaches  f"  the  direction  cosines  of  the  tangent  to  the 


FUNDAMENTAL   EXISTENCE   THEOREMS.  65 

curve  (22),  for  increasing  z,  approach  the  values 


on  one  of  the  arcs  PiP2  and  P^Pz',  on  the  other  the  limiting 
direction  is  exactly  the  opposite,  since  the  values  of  D  on  the 
two  arcs  have  opposite  signs.  Hence  if  w  =  z  increases  along 
the  arc  PiP2  it  must  decrease  along  P2Ps,  and  vice  versa. 

In  the  xy-p\a,ne  these  results  mean  that  the  images  of  the 
arcs  PiP2  and  P2Ps  are  two  arcs  of  the  circle  r  =  k  which  overlap 
near  the  image  of  P2;  the  images  of  S  and  S'  must  therefore 
be  superposed  in  the  vicinity  of  the  image  of  OE^. 

If  the  boundary  OE2  between  S  and  S'  is  not  one  of  the  elements 
of  D,  the  images  of  the  two  regions  in  the  xy-p\ane  will  adjoin 
each  other  along  the  image  of  OE2,  and  the  inverse  functions 
(20)  will  be  analytic  at  every  point  of  the  image  of  OE2  except 
the  origin.  For  at  such  points  the  functional  determinant  D  is 
different  from  zero. 

By  combining  the  results  which  have  so  far  been  deduced, 
the  truth  of  the  following  theorem  is  established: 

For  a  transformation 

(23)  x  =  <p(u,  v),        y  =  \f/(u,  v) 

with  the  characteristics  described  in  the  first  paragraph  of  this 
section,  a  circle  C  can  be  selected  in  the  uv-plane  with  center  at  the 
origin  and  having  the  following  properties:  The  circle  is  inter- 
sected by  each  real  element  of  the  functional  determinant  D  at  some 
first  point  P.  The  arcs  OP  so  determined  on  the  different  elements 
divide  the  interior  of  C  into  regions  Si,  <S2,  •  •  • ,  Sk.  The  points 
of  each  region  S  correspond  in  a  one-to-one  way  by  means  of 
equations  (23)  with  the  points  of  a  sheet  S  of  the  xy-plane  which 
winds  about  the  origin  and  is  bounded  by  the  images  of  the  bound- 
aries of  S.  The  single-valued  functions 

(24)  u  =  f(x,  y),        v  =  g(x,  y) 
6 


66 


THE  PRINCETON  COLLOQUIUM. 


so  determined  are  continuous  at  all  points  of  the  sheet  2  and  analytic 
in  the  interior  of  2.  //  in  two  adjoining  regions,  say  Si  and  82, 
the  signs  of  D  are  opposite,  then  the  images  2i  and  22  overlap  in 
the  neighborhood  of  their  common  boundary  OTT<> ;  if  the  signs  of  D 
are  the  same,  the  regions  2i  and  22  adjoin  along  07r2  without  over- 
lapping. 

The  adjoining  figure  illustrates  the  case  when  D  has  four  real 
elements  and  the  signs  of  D  are  opposite  in  any  two  adjoining 
regions  S.  Further  illustrations  of  the  theorem  are  given  in  §  14. 


FIG.  6. 


It  has  not  been  proved  above  that  the  functions  (24)  are 
continuous  on  a  boundary  OTT  of  one  of  the  regions  2.  Suppose 
that  TT  is  a  point  of  such  a  boundary,  and  let 


(25) 


l>    7T2, 


be  any  sequence  of  points  of  2  with  limit  TT.     The  corresponding 
points 

(26)  pi,  pi,  p3,  •" 

of  S  have  condensation  points  in  S,  one  of  which  may  be  denoted 
by  p.     There  is  then  a  sub-sequence 


Pi 


PS 


FUNDAMENTAL   EXISTENCE   THEOREMS.  67 

among  the  points  (26)  whose  limit  is  p;  and  on  account  of  the 
continuity  of  the  functions  (23),  the  corresponding  points 

(27)  7T/,    7T2',    7T3',     •  •  • 

of  the  sequence  (25)  must  have  as  limit  point  the  image  of  p 
in  S.  But  the  limit  of  (27)  is  necessarily  TT,  and  TT  is  therefore 
the  image  of  p.  It  follows  at  once  that  the  sequence  (26)  has 
a  unique  limit  point  p  which  is  the  image  of  IT,  and  from  this 
property  the  continuity  of  the  functions  (24)  in  the  ordinary 
sense  can  be  readily  deduced. 

The  functions  <f>,  \f/,  and  D  can  be  expanded  in  the  form 


<P  =•    <Pm  ~ 
(28)  $  =  \f/n  +  *f/n+l  +    '  *  ' , 

D  =  Z)m+n_2  ~r  Dm+n—  i  T"  •  •  • , 

where  <pk,  ^k,  Dk  are  homogeneous  polynomials  in  u,  v  of  degree. 
Jc,  and 


du      dv 

Dm+n-2  =       ~  ,  ~  . 

Oy/n       d\f/n 

du       dv 

If  the  real  roots  of  (pm,  \f/n,  and  Dm+n-i  are  all  simple  roots  and 
distinct  from  each  other,  there  will  be  an  element  of  <p,  $,  or  D 
in  each  of  the  corresponding  directions,  and  a  notion  of  the 
character  of  the  transformation  can  be  derived  without  difficulty. 
In  the  applications  of  §  14  this  remark  is  of  frequent  service. 

§  12.    THE  CASE  WHERE  THE  FUNCTIONAL  DETERMINANT 
VANISHES  IDENTICALLY 

It  is  well  known  that  when  the  functional  determinant  of 
two  analytic  functions  <p  and  \f/  vanishes  identically,  then  near 
any  point  where  not  all  of  the  derivatives  <pu,  <pv,  $u,  4/v  vanish 
the  functions  <p  and  \(/  satisfy  a  relation  of  the  form 

F(<p,  ,/>)  =  0 


68  THE   PRINCETON   COLLOQUIUM. 

identically  in  u  and  v.  It  is  possible  to  show  that  such  a  relation 
exists  also  near  a  singular  point  at  which  the  four  derivatives 
above  all  vanish. 

If  a  relation  can  be  found  after  a  substitution  of  the  form 

u  =  mil  +  |3i'i,        v  =  yui  +  8vi, 


for  which  a8  —  /3y  does  not  vanish,  then  it  will  surely  be  satisfied 
when  MI  and  v\  are  replaced  by  the  original  variables  u,  v. 

Suppose  then  that  the  analytic  functions  <p  and  \f/  have  already 
been  prepared  by  a  transformation  (29)  in  such  a  way  that  in 
the  expansions  (28)  <pm  and  \[<n  both  contain  terms  in  u  alone. 
By  applying  the  preparation  theorem  of  Weierstrass  to  the 
functions  <p(u,  v)  —  x  and  \f/(u,  v)  —  y,  two  polynomials 


P(u,  v,  x)  =  um  +  aium~l  +  •  •  •  H-  a 
Q(u,  v,  y}  =  un  +  blUn~l  +  ----  h  bn 


m, 


B,re  obtained,  whose  coefficients  are  convergent  series,  without 
constant  terms,  in  v,  x  and  v,  y,  respectively.  In  a  certain  vicinity 

X    <    €,         y     <    €,        U     <    €,         V     <    € 

the  only  solutions  of  the  equations 

(30)  <p(u,  v)  -  x  =  0,        $(u,  v)  -  y  =  0 

are  values  (u,  v,  x,  y)  which  make  P  and  Q  vanish  also,  and  vice 
versa. 

The  resultant  of  P  and  Q  is  a  convergent  series  R(v,  x,  y) 
for  which  R(Q,  x,  y)  does  not  vanish  identically.  For  if  all  of 
the  coefficients  of  R(0,  x,  y)  were  zero,  there  would  be  a  region 

(31)  «  =  0,      x  <  5,     \y\  <  5  (5^«) 

at  any  point  of  which  the  polynomials  P  and  Q  have  a  common 
root  in  absolute  value  less  than  e,  and  the  set  of  values  (u,  0,  x,  y) 
so  defined  satisfies  also  the  equations  (30).  The  existence  of 
such  a  region  is,  however,  impossible,  since  when  y'  is  given 


FUNDAMENTAL  EXISTENCE  THEOREMS.  69 

satisfying  (31),  a  value  x'  can  always  be  selected  which  is  dif- 
ferent from  the  values  of  <p(u,  0)  at  all  of  the  n  roots  of  Q(u,  0,  y'). 
For  such  a  set  v  =  0,  x',  y'  in  the  region  (31)  there  would  be  no 
corresponding  value  u'  satisfying  the  equations  (30). 

The  resultant  R(v,  x,  y)  vanishes  identically  in  u,  v  when  x 
and  y  are  replaced  by  <p  and  \f/.  For  R  is  expressible  in  the  form 

R(v,  x,  y)  =  MP  +  NQ, 

where  M  and  N  are  polynomials  in  u  with  coefficients  which  are 
series  in  v,  x,  y,  and  P  and  Q  vanish  identically  when  x  =  <p, 

y  =  t- 

The  series  R(0,  <p,  ^)  vanishes  identically  in  u,  v.  If  not,  there 
would  be  a  straight  fine  u  =  kv  on  which  R(0,  <p,  ^)  and  <pu 
are  different  from  zero  except  at  the  origin.  Let  (uf,  v'}  be  a 
point  of  this  line  near  (u,  v)  =  (0,  0),  at  which  <p  and  \f/  have  the 
values  <p'  and  \f/',  respectively.  The  series 

(32)  R(Q,  <p,  *)  +  fl.(0,  ?,*)«+••• 
vanishes  identically,  in  particular  along  the  curve 

(33)  <p(u,  v)  =  <p' 

through  the  point  (uf,  vf).  Since  <pu  does  not  vanish  at  (u1  ',  «'), 
this  curve  can  be  expressed  in  the  form 

u  =  U(v), 
and  along  it 

4«17,  .)-(#.  -ft  +  *]  =0, 

<*V  L  <Pu  Ju=U(v) 

since  the  functional  determinant  of  <p  and  \l/  vanishes  identically. 
On  the  curve  (33)  the  function  \f/  has  therefore  the  constant 
value  \l/',  and  the  series  (32)  takes  the  form 


and  vanishes  identically  in  v.     Its  coefficients  must  therefore 
all  vanish,  since  a  series  whose  zeros  have  a  point  of  condensation 


70  THE  PRINCETON  COLLOQUIUM. 

in  the  interior  of  its  circle  of  convergence  must  have  all  of  its 
coefficients  equal  to  zero.  This  contradicts,  however,  the  as- 
sumption that  a  point  (uf,  vr)  exists  at  which  R(0,  <p,  \f/)  does  not 
vanish. 

It  has  been  shown  therefore  that  in  case  the  functional  deter- 
minant of  the  two  convergent  series 

<P  =   <Pm  +  <Pm+l  +   '  '  *  i 


vanishes  identically,  the  two  functions  <p,  \f/  satisfy  a  relation  of  the 
form 

)  =  0 


identically  in  u,  v,  where  F  is  itself  a  convergent  series  in  its  two 
arguments.  This  statement  is  true  even  when  <p  and  \f/  both  have 
singular  points  at  the  origin. 

It  is  evident  that  when  D  =  0  the  transformation 

x  =  <p(u,  v),        y  =  $(u,  v) 

makes  all  of  the  points  in  the  neighborhood  of  the  origin  in  the 
wfl-plane  correspond  to  points  on  the  various  branches  of  the 
curve 

F(z,  y)  =  0 

in  the  ay-plane.  The  points  (x,  y)  which  are  obtained  by  the 
transformation  do  not  cover  any  region. 

§  13.    A  GENERALIZATION  OF  THE  PREPARATION  THEOREM  OF 

WEIERSTRASS 

Consider  for  a  moment  two  functions 
(34)          f(u,  v,  xlt  x2,  •  ••,  Xm),     g(u,  v,  Xi,  x2,  •••,£»,) 

which  are  polynomials  in  the  variables  u,  v  and  have  for  coefficients 
convergent  series  in  x\t  xz,  •  •  •  ,  xm.  According  to  the  usual 
algebraic  theory  of  elimination,  there  exists  a  polynomial  p  in  v 


FUNDAMENTAL   EXISTENCE   THEOREMS.  71 

which  has  convergent  series  in  the  x's  as  coefficients,  and  which 
is  linearly  expressible  in  the  form 

p  =  cf  +  dg, 

where  c  and  d  are  polynomials  of  the  same  character  as  /  and  g. 
If  a  set  of  variables  (u,  v,  x)  make  /  and  g  both  vanish,  then  v 
must  be  a  root  of  the  polynomial  p;  and  conversely  to  any  root 
of  p  corresponding  to  given  values  x,  there  exists  at  least  one 
pair  of  values  (u,  v)  which  satisfy  the  two  equations  /  =  g  =  0. 

There  is  a  generalization  of  the  preparation  theorem  of  Weier- 
strass  from  which  similar  results  may  be  deduced  with  respect 
to  two  functions  /  and  g  which  are  not  polynomials  but  series  in 
the  variables  u  and  v,  and  with  respect  to  the  roots  of  such 
functions  in  a  neighborhood  of  any  set  of  values  (u0,  VQ,  z0) 
making  /  and  g  vanish.  As  in  the  proof  of  the  theorem  of  §  9, 
the  point  in  whose  neighborhood  /  and  g  are  to  be  studied  may 
be  taken  without  loss  of  generality  at  the  origin. 

Suppose  then  that  f  and  g  are  two  convergent  series  in  u,  v,  x 
vanishing  for  (u,  v,  x)  =  (0,  0,  0),  and  such  thatf(u,  v,  0,  0,  •  •  •,  0) 
and  g(u,  v,  0,  0,  •••,0)  have  no  common  factor.  Then  there 
exists  a  polynomial 

(35)  p  =  vn  +  pie*-1  + h  pn, 

in  which  the  coefficients  pk  (k  =  1,  2,  •  •  •,  n)  are  convergent  series 
in  x  having  no  constant  terms,  with  the  following  properties:  (1)  it 
is  linearly  expressible  in  the  form 

p  =  cf  +  dg, 

where  c  and  d  are  convergent  power  series  in  u,  v,  x;  (2)  in  a  properly 
chosen  neighborhood 

(36)  u  <  e,     \v  <.  €,     \x  <  e 

every  root  (u,  v,  x)  of  f  and  g  must  also  make  p  vanish;  (3)  there 
exists  a  constant  d  ^  e  such  that  for  any  x  in  the  region 

(37)  \x\  <  d 


72  THE   PRINCETON   COLLOQUIUM. 

there  is  associated  with  each  root  v  of  p  a  solution  (u,  v,  x)  of  the 
equations  f  =  g  =  0  satisfying  the  inequalities  (36).* 

If/(w,  v,  0,  0,  •  •  •,  0)  and  g(u,  v,  0,  0,  •  •  •,  0)  have  no  common 
factor,  then  one  at  least  of  them,  say  /,  has  terms  in  the  variable 
u  alone,  and  according  to  the  preparation  theorem  of  Weier- 
strass  f(u,  v,  x)  has  as  factor  a  polynomial  of  the  form 

(38)  a0um  +  a^-1  +  ----  h  «m-iw  +  am  =  &/, 

in  which  a0  is  a  constant  different  from  zero,  and  01,  a2,  •  •  •  ,  am 
are  series  in  v,  x  without  constant  terms.  The  symmetric 
functions  of  the  roots  u\,  w2,  •  •  •  ,  um  of  this  polynomial  are 
expressible  rationally  and  integrally  in  terms  of  the  coefficients 
o>\,  02,  '  •  •  ,  am,  and  are  therefore  convergent  series  in  v,  x.  The 
product 


(39)  ffot,  »,  *)  =  *(»,  *) 

*=i 

is  a  convergent  series  in  u^,  v,  x,  also  symmetric  in  the  variables 
Wfc,  and  hence  expressible  as  convergent  series  in  v,  x. 

The  function  h(v,  0)  does  not  vanish  identically,  on  account  of 
the  hypothesis  that  f(u,  v,  0,  0,  •  •  •,  0)  and  g(u,  v,  0,  0,  •  •  •,  0) 
have  no  common  factor.  If  it  did  vanish  identically,  then  for 
every  sufficiently  small  value  of  v  one  at  least  of  the  expressions 
g(uk,  v,  0)  would  vanish.  But  in  §  10  it  was  seen  that  when 
f(u,  v,  0)  and  g(u,  n,  0)  have  no  factor  in  common,  there  is  always 
an  interval  0  <  v  ^  v\  in  which  there  is  no  value  v  belonging  to 
a  pair  (u,  0)  making  both  of  these  functions  vanish. 

The  preparation  theorem  of  Weierstrass  can  therefore  be 
applied  also  to  the  function  h(v,  x),  and  the  polynomial  so  found 
is  the  one  desired  in  the  theorem.  For,  in  the  first  place,  a  constant 
«  can  be  chosen  so  small  that  every  root  (u,  v,  x)  of  /  and  g  in 
the  region  (36)  must  be  one  of  the  sets  (uk,  v,  x),  and  must  make 

*  A  proof  that  the  values  of  u  and  v  belonging  to  the  roots  of  a  system  of 
equations  of  the  form  (34)  are  roots  of  polynomials  similar  to  (35)  was  given 
by  Poincar6  in  the  introduction  to  his  Thesis,  "  Sur  les  proprie'te's  des  fonctions 
d^finies  par  les  Equations  aux  differences  partielles,"  Paris  (1879). 


FUNDAMENTAL  EXISTENCE  THEOREMS.  73 

the  product  (39),  and  hence  p,  vanish.  In  the  second  place,  a 
constant  5  ^  e  can  be  taken  so  small  that  every  root  v  of  p  as 
well  as  the  corresponding  sets  (uk,  v,  x)  lie  in  the  domain  (36). 
One  at  least  of  these  sets  must  evidently  satisfy  g  =  0  as  well  as 
/  =  0.  The  restrictions  on  5  and  e  have  been  stated  somewhat 
roughly,  but  the  reader  will  readily  convince  himself  that  these 
quantities  may  be  selected  so  that  the  convergence  of  the  different 
series  and  their  equivalence  with  the  corresponding  polynomials 
are  properly  adjusted. 

Finally,  the  polynomial  p  is  linearly  expressible  in  the  form 
described  in  the  theorem,  in  terms  of  /  and  g.  To  prove  this, 
suppose  that  the  above  process  has  been  applied  to  the  functions 
/  —  a  and  g  —  /3.  A  polynomial  P(v,  x,  a,  /3)  with  coefficients 
which  are  series  in  x,  a,  /3  is  then  found,  which  may  be  written 
in  the  form 

P(v,  x,  a,  0)  =  P(v,  x,  0,  0)  +  Ca  +  Z>/3, 

where  C  and  D  are  convergent  series  in  the  arguments  of  P. 
The  series  P(u,  x,  f,  g)  vanishes  identically  in  u,  v,  x  since  P  =  0 
must  be  satisfied  by  every  set  of  variables  (u,  v,  x,  a,  /3)  in  a 
neighborhood  of  the  origin  which  make  /  —  a  and  g  —  0  vanish, 
certainly  then  by  the  set  (u,  v,  x,f,  g).  Hence 

P(t>,  x,  0,  0)  =  -  Cf  -  Dg 

is  an  identity  in  u,  v,  x,  when  a  and  |8  are  replaced  in  C  and  D 
by  the  series/,  g.  But  P(v,  x,  0,  0)  is  precisely  the  polynomial 
p(v,  x)  found  above,  since  for  a  =  0  =  0  the  steps  in  the  con- 
struction of  P(v,  x,  0,  0)  are  identical  with  those  used  in  finding  p. 
If  the  series  f(u,  v,  0,  0,  •  •  •,  0)  and  g(u,  v,  0,  0,  •  •  •,  0)  begin 
with  homogeneous  polynomials  having  no  common  factor  of  degrees 
m  and  n,  respectively,  then  the  degree  of  the  polynomial  pis  v=  mn.* 

*  In  a  paper  of  recent  date  the  writer  has  developed  a  generalization  of 
this  theorem  and  the  results  which  follow,  for  a  system  of  equations  of  the  form 
fi(xi,  x2,  ••  -,xm;  ?/i,  7/2,  •  •  •,  yn)  =  0  (i  =  1,  2,  •  •  •,  n).  See  Transactions  of 
the  American  Mathematical  Society,  vol.  13  (1912),  p.  133. 


74  THE   PRINCETON   COLLOQUIUM. 

Let  the  lowest  terms  of  f(u,  v,  0,0,  •  •  •  ,  0)  and  g(u,  r,  0,  0,  •  •  •  ,  0) 
be  denoted  by  <pm(u,  v)  and  \j/n(u,  v),  respectively.  One  of  the 
two,  say  <f>m,  has  a  term  involving  u  alone  with  coefficient  different 
from  zero,  since  <pm  and  \fsn  have  no  common  factor.  The  terms 
of  lowest  degree  in  the  polynomial  (35)  are  also  (pm,  since  the 
series  b  has  constant  term  unity.  In  the  product  (39)  the  terms 
may  be  rearranged  into  groups  of  the  form  cvpU,  where  U  is  a 
homogeneous  symmetric  function  of  a  certain  degree  a  in 
u>i,  Uz,  •  •  •  ,  um.  The  expression  for  such  a  symmetric  function 
is  isobaric  and  has  the  weight  a  in  the  coefficients  of  the  poly- 
nomial (35).  When  x  =  0  the  terms  of  lowest  degree  in  U  will 
be  at  least  of  degree  a  in  v,  since  each  coefficient  a*  of  (35)  begins 
with  the  coefficient  of  um~k  in  the  polynomial  (pm(u,  v).  The 
terms  of  lowest  degree  in  v  alone  in  the  product  (39)  will  there- 
fore be  those  of  the  product 


and  they  have  the  value  vmnR/ao,  in  which  a0  is  the  coefficient  of 
vm  in  <f>m(u,  v)  and  R  is  the  resultant  of  <pm(\,  v)  and  \f/n(l,  v).* 
But  since  <pm  and  \f/n  have  no  common  factor  the  coefficient  of 
fl*1"  is  surely  different  from  zero,  and  the  theorem  last  stated 
follows  at  once. 
//  the  substitution 

v  =  —  tu  -\-  z 

is  made,  in  which  t  is  a  new  variable,  the  series 

F(u,  z,  x,  t)  =  f(u,  z  —  tu,  x), 
G(u,  z,  x,  t)  =  g(u,z  —  tu,  x) 

have  a  polynomial 

(41)  P(z;x,  t)  =  CF+  DG 

with  properties  similar  to  those  of  p  and  of  the  same  degree  v.     In 

*  See,  for  example,  Konig,  Einleitung  in  die  allgemeine  Theorie  der  alge- 
braischen  Grossen,  p.  311  and  p.  271  (d). 


FUNDAMENTAL  EXISTENCE   THEOREMS.  75 

a  properly  chosen  region 

(42)  |w|  <  e,      v    <  e,      x    <  € 

every  root  (u,  v,  x)  of  f  and  g  defines  a  factor  z  —  tu  —  v  of  P. 
If  d  ^  6  is  sufficiently  small  and  x  a  set  of  variables  satisfying 

(43)  x  <  d, 

then  P  has  v  factors  of  the  form  z  —  tu  —  v,  for  each  of  which  the 
values  (u,  v,  x)  are  a  solution  of  the  equations  f  =  g  =  0  in  the 
region  (42). 

The  degree  of  P  must  be  the  same  as  that  of  p,  since  for 
x  =  t  =  0  the  series  F(u,  z,  0,  0),  G(u,  z,  0,  0)  are  identically 
equal  to  the  series /(w,  v,  0)  and  g(u,  v,  0)  when  v  is  replaced  by  z. 
In  a  certain  region 

(44)  \u   <  ei,     |z|  <  ei,      .T   <  ei,     \t\  <  «i, 

where  ei  is  for  convenience  taken  less  than  unity,  every  root 
system  (u,  z,  x,  t)  of  F  and  G  makes  P  vanish  also.  If  e  is  taken 
less  than  ei/2  and  t  is  restricted  to  the  range  \t\  <  ci,  every  root 
system  (u,  v,  x)  of  /  and  g  in  the  region  (42)  gives  values  u, 
z  =  tu  +  v,  x,  t  satisfying  the  inequalities  (44),  and  hence  P 
must  vanish  identically  in  t  and  have  z  —  tu  —  v  as  a  factor. 

Suppose  then  that  e  is  a  constant  satisfying  the  requirements 
of  the  theorem  with  respect  to  the  region  (42),  and  that  the  region 
analogous  to  (37)  for  the  polynomial  P  and  the  constant  e/2  is 

(45)  \x  <  d,         \t\<  5; 

and  let  x  =  %  be  any  set  of  values  satisfying  these  inequalities. 
If  the  discriminant  of  P  is  not  identically  zero  in  t  for  x  =  £, 
a  value  t  =  r  can  be  selected  also  satisfying  (45)  and  such  that 
all  the  roots  z  of  P  corresponding  to  the  values  £,  r  are  distinct. 
There  are  then  v  distinct  root  systems  (u,  z,  £,  T)  satisfying  the 
inequalities  (41)  with  ei  replaced  by  e/2.  The  corresponding 
values  (u,  v  =  z  —  tu,  £)  are  v  distinct  roots  of  /  and  g  lying  in 
the  region  (41).  According  to  the  paragraph  just  preceding,  P 
has  therefore  v  distinct  factors  z  —  tu  —  v. 


76  THE   PRINCETON    COLLOQUIUM. 

In  case  the  discriminant  of  P  vanishes  identically  in  t  for  x  =  £, 
the  multiple  factors  of  P(z;  £,  0  can  be  separated  out  by  the 
highest  common  divisor  process,  and  the  factorization  of  the 
resulting  polynomial  can  then  be  discussed  in  a  manner  similar  to 
that  just  explained.  In  either  case,  therefore,  P(z;  £,  t)  has  only 
linear  factors  of  the  form  z  —  tu  —  v. 

The  number  and  character  of  the  root  systems  (u,  v,  x)  of  the 
functions  /  and  g  in  the  neighborhood  of  the  origin  are  well  de- 
fined by  means  of  the  polynomial  P(z;  x,  t).  To  any  x  in  the 
region  (43)  there  correspond  v  root  systems  (u,  v,  x)  not  neces- 
sarily all  distinct,  and  the  ^-valued  functions  u(x),  v(x)  so  defined 
are  continuous.  This  is  evidently  true  for  the  function  v(x), 
since  its  values  are  the  roots  of  the  polynomial  P(v;  x,  0)  whose 
coefficients  are  analytic  in  x.  Similarly  z  is  continuous  in  x,  t, 
since  its  values  are  the  roots  of  P(z;  x,  t},  and  it  follows  that 
u  =  (z  —  v)/t,  for  a  fixed  value  t  =t=  0,  must  be  continuous  in  x. 

If  P  is  not  irreducible,  that  is,  not  decomposable  into  similar 
factors  of  lower  degrees,  its  discriminant  A  (a-,  t)  can  not  vanish 
identically  in  x,  t.  At  any  value  x  =  £  where  A(£,  t)  is  not 
identically  zero  in  t,  the  v  factors  z  —  tu  —  v  of  P  are  all  distinct. 
If  t  =  T  is  selected  so  that  A(£,  T)  4=  0,  the  roots  of  P  are  distinct 
analytic  functions  of  x  and  t  in  the  neighborhood  of  £,  T,  and 
the  corresponding  values  of  u  and  v  are  analytic  functions  of  x 
in  the  neighborhood  of  £. 

The  values  x  =  %  near  which  the  ^-valued  functions  u,  v  do 
not  surely  have  v  distinct  analytic  branches,  are  those  for  which 
A(£,  f)  vanishes  identically  in  t.  At  such  a  point  some  of  the 
values  of  the  root-systems  (u,  v)  coincide,  and  only  those  which 
are  distinct  belong  necessarily  to  analytic  branches  of  the 
functions  u,  v.  The  values  £  which  make  A(£,  f)  identically 
zero  must  belong  to  one  of  the  totalities  of  points  defined  by 
equating  to  zero  the  coefficients  of  the  finite  number  of  powers 
of  t  in  the  discriminant  A  (a:,  /).* 

*  For  the  characterization  of  these  totalities  after  the  method  of  Kronecker 
for  algebraic  equations,  see  Kistler,  "Ueber  Funktionen  von  mehreren  komplexen 
Veranderlichen,"  Dissertation,  Gottingen,  1905. 


FUNDAMENTAL  EXISTENCE  THEOREMS. 


77 


If  P(z,  x,  t)  is  reducible,  arguments  similar  to  those  above 
can  be  applied  to  any  one  of  its  irreducible  factors. 

The  multiple  roots  (u,  v,  x)  of  the  functions  f  and  g  are  character- 
ized by  the  property  that  the  functional  determinant  d(f,  g)fd(u,  v) 
is  zero  at  such  points. 

For  from  the  identity  (41)  in  u,  z,  x,  t,  it  follows  by  differ- 
entiation that 

0  s  CUF  +  DUG  +  CFU  +  DGU, 
P2  EE  CZF  +  DZG  +  CF,  +  DG2. 


If  the  determinant 


Fu    F,--fu 

r*       r< 

(JM      trJ 


vanishes  at  a  solution  (u,  v,  x)  of  /  =  g  =  0,  the  two  equations 
above  show  that 

CFU  +  DGU  =  0,        Pz  =  CFZ  +  DGZ  =  0 

for  the  values  (u,  z  =  —  tu  +  v,  x);  and  it  follows  that  z—tu—v 
is  a  multiple  factor  of  P,  since  it  occurs  also  in  P2. 

On  the  other  hand  suppose  that  at  a  set  of  values  (u',  v',  £) 
the  determinant  d(f,  g)/d(u,  v)  is  different  from  zero,  while/  and 
g  vanish.  It  is  to  be  shown  that  the  polynomial  P(z;  £,  t)  has 
tu'  +  v'  as  a  simple  root.  All  of  the  roots  of  P(z;  £,  t)  have  the 
form  tu  +  v}  and  some  are  perhaps  multiple.  Those  which  are 
distinct  will  remain  distinct  for  a  numerical  value  t  =  T  if  T 
is  properly  selected,  and  the  derivative 


(47) 


Fu(u',  r,  S,  T)  =  /„(«',  «',  £)  - 


can  at  the  same  time  be  made  different  from  zero,  £"  being  the 
expression  TU'  +  v'.     In  the  expressions 

(48)  A0um  +  ^iw"-1  +  ----  h  4m-iM  +  An  =  BF, 

m 

(49)  T[G(uk,z,  £,T)  =  H(z,  €,  r), 


78  THE   PRINCETON   COLLOQUIUM. 

analogous  to  (38)  and  (39)  for  the  functions  F(u,  z,  £,  t)  and 
G(u,  z,  £,  t),  the  factor  G(u\,  z,  £,  T),  where  u\  is  the  root  of  (48) 
which  reduces  to  u'  for  z  =  £,  is  the  only  one  which  vanishes 
for  2  =  f .  To  prove  this  it  can  be  seen  in  the  first  place  that 
u'  is  a  simple  root  of  (48)  for  z  =  £" ,  since  the  derivative  (47)  is 
different  from  zero.  Furthermore  when  z  =  f  no  other  root  u?. 
distinct  from  u\  can  make  G(UZ,  z,  £,  T)  vanish.  Otherwise  / 
and  g  would  vanish  not  only  at  the  values  (uf,  vr,  £),  but  also  at 
(uz,  £  —  ruz,  £),  where  uz'  is  the  value  of  «2  for  z  =  £";  and 
P(z;  £,  f)  would  have  two  roots,  tu'  +  v'  =  tu'  +  f  —  ru'  and 
tuz  +  f  —  rw2',  which  are  distinct  for  <  4=  T  and  equal  to  $" 
when  2  =  r.  On  account  of  the  way  in  which  T  was  selected, 
this  is  impossible. 

The  root  u\  of  (48),  that  is  to  say  also  of  F,  has  an  expansion 
of  the  form 


/     *\7O7*»7"//  >*\_1 

F  (uf  f  £  T) 
in  powers  of  z  —  £";  and  the  value  of  G(u\,  z,  £,  T)  is  a  series 

p   n  r>  ri 

-FulJz  —   f  2lJu 

p ~  (2          D  "T    *  '  * 
r  u 

whose  first  term  is  different  from  zero,  since  for  the  values 
(u',  f ,  £,  r)  we  have 

"    r\  =  \*n(?J  I'  S    {*/«/  !''  tv  *  °' 

*Ju     vj«;       >Qu\u  »  ®  j  sJ     fl'vi.w  j  ^  >  ?/ 

as  is  readily  seen  from  equations  (40).  Hence  the  quotient 
H(z,  £,  T)/(Z  —  f)  is  different  from  zero,  and  neither  H(z,  £,  t) 
nor  its  polynomial  P(z;  ^,  0  can  have  more  than  one  factor 
2  -  tu'  -  v'. 

§  14.    APPLICATIONS  OF  THE  PRECEDING  THEORY 
The  real  transformation 


.  x  =  <f>(u,  n)  =  aiou  +  ooifl  +  azoir  + 

y  =  \l/(u,  v)  =  biou  +  b0iv  +  620w2  + 


FUNDAMENTAL   EXISTENCE   THEOREMS.  79 

has  a  singular  point  at  the  origin  when 


(51) 


=  0. 

|010       001 


If  one  of  the  elements  of  the  determinant  is  different  from  zero, 
it  may  be  assumed  without  loss  of  generality  to  be  a™;  then 
after  two  transformations 


V    =  V, 

,  ,  bw 

x  -  x,  y  =  —  --  x+  y 

«io 

the  equations  (50)  take  the  form 

x  =  u  -j-  azou2  -f  a\\uv  +  aoa*2  +  •  •  •  , 


(52) 

y  =  bzoU2  +  bnuv  +  602«2  + 


For  convenience  the  primes  have  been  dropped,  and  the  notation 
for  coefficients  of  terms  of  higher  degree  than  the  first  is  the  same 
as  that  in  the  original  equation.  It  may  further  be  supposed 
that  the  polynomials 


<f>i  =  u,        \f/2  =  bzov?  +  bnuv  +  &0202 

have  no  common  factor,  in  other  words  that  602  ={=  0.     The  origin 
is  then  a  singular  point  for  the  transformation  (50)  of  a  very 
general  type,  since  aside  from  the  assumption  (51)  only  inequalities 
on  the  coefficients  of  the  series  have  been  exacted. 
The  functional  determinant  has  the  expansion 

D(u,  v)  =  bnu  +  2&020  +  •  •  -i 
and  hence  has  a  single  branch 

611 

v  =  —  -,    u  +  •  •  •  , 

&02 

along  which  D  vanishes  and  on  opposite  sides  of  which  D  has 
different  signs.     The  image  A  of  this  curve  in  the  zy-plane  has 


80 


THE   PRINCETON   COLLOQUIUM. 


an  ordinary  point  at  the  origin,  as  shown  by  its  equations 
x=  u+  •-,        y  =  - 


-bn2 


'OJ 


The  region  S  in  the  figure  has  in  it  one  real  element  of  <p  and  at 
most  two  of  \f/,  since  the  solutions  of  <p  =  0  lie  on  a  single  real 


FIG.  7. 

curve  through  the  origin,  and  those  of  ^  =  0  are  either  imaginary 
or  else  lie  on  two  real  branches.  Hence  the  region  S  which  is  the 
image  of  S  lies  on  one  side  only  of  the  curve  A  and  overlaps  the 
image  2'  of  S'. 

Since  <p\  and  \f/2  have  no  common  factor,  the  theorems  of  §  13 
show  that  there  exist  two  constants,  8  and  e,  such  that  the  equa- 
tions (52)  have  two  and  only  two  solutions  [u\(x,  y),  v\(x,  y),  x,  y], 
[uz(x,  y),  vi(x,  y),  x,  y]  in  the  region 


Ittl  <  €,         v  <  e, 


x  <  e, 


\y\< 


corresponding  to  any  (x,  y)  in  the  region 

\x  <  5,         \y\  <  8. 

The  functions  u\,  Vi,  u^,  02  so  defined  are  everywhere  continuous 
and  the  two  solutions  above  are  analytic  and  distinct  except 
along  the  curve  A.  On  one  side  of  A  they  are  imaginary,  on 
the  other  real. 


FUNDAMENTAL   EXISTENCE   THEOREMS. 


81 


Another  interesting  case  is  that  of  a  transformation  (50)  for 
which  again  the  coefficients  are  real,  and 


d<p  _  d\(/ 
du~  dv' 


d<f> 
dv 


du' 


Such  a  transformation  might  be  called  a  monogenic  transforma- 
tion. It  follows  at  once  that  <p  and  \f/  must  begin  with  two 
homogeneous  polynomials,  <pm  and  \f/m,  of  the  same  degree  ra, 
which  also  satisfy  the  last  equations.  Consequently 

<pm  +  tym  =  (a  +  ib)  (u  +  iv)m  =  Pm(a  +  ib)  (cos  6  +  i  sin  B}m 
and 
<pm  =  Pw(«  cos  mO  —  b  sin  md),    \f/m  =  pm(a  sin  md  +  b  cos  md), 

where  a  and  b  are  not  both  zero.  These  equations  show  that 
<f>m(u,  v)  and  \f/m(u,  v)  have  each  m  real  linear  factors  in  u,  v, 
and  that  no  factor  of  <pm  is  also  in  \l/m. 

The  determinant  D(u,  v)  has  an  expansion 


where 


D(u,  r)  = 


-i  + 


du      dv 
dv 


du 


' 


dv 


du 


The  homogeneous  polynomial  D^m-i  has  no  real  root,  since  such 
a  root  would  necessarily  belong  to  both  d<pmfdu  and  d<pm/dv,  and 
from  the  equations 


m<pm=u 


du 


=  —  u 


d<pm  difrn 

dv  du 


it  follows  that  <pm  and  \f/m  would  then  have  a  common  factor. 
Hence  there  are  no  real  points  at  which  D  vanishes  near  the 
origin  in  the  wa-plane. 

7 


82 


THE  PRINCETON  COLLOQUIUM. 


The  argument  of  §  11  shows  that  the  elements  of  <pm  and  \J/m 
separate  each  other  and  that  a  neighborhood  of  the  origin  in 
the  wfl-plane  is  transformed  into  a  sheet  winding  m  times  around 
the  origin  in  the  a^-plane,  as  shown  in  the  figure.  This  is  the 


well-known  transformation  of  the  neighborhood  of  the  origin 
in  a  complex  w-plane  by  means  of  a  relation  of  the  form 

2  =  Awm  +  A'wm+l  +  •  •  •  i 

where  z  =  x  -f-  iy  and  w  =  u  -\-  iv.     The  figure  is  drawn  for  m  =  3. 

There  are  many  other  special  cases  similar  to  those  just  given 
which  might  be  elucidated  by  means  of  the  theorems  of  the 
preceding  sections,  but  for  which  the  methods  in  the  two  ex- 
amples just  given  are  typical.  It  may  be  of  interest,  however, 
to  exhibit  an  example  which  illustrates  the  use  of  the  theorems 
of  §  8,  as  well  as  the  behavior  of  a  transformation  at  singular 
points. 

Suppose  that  the  real  i/0-plane  is  transformed  by  means  of 
the  equations 


(53) 


x  =  -r  —  uv  +  „  +  ^-, 

^  j£          O 


=  T  +  uv  + 


02 


FUNDAMENTAL  EXISTENCE  THEOREMS. 

The  functional  determinant  has  the  value 

D(u,  v)  =  (u  H-  v)(uz  -\-2u-  20) 
and  it  vanishes  along  the  curves 

t)  =  —  u,        v  =  u  +  TT. 


83 


which  have,  respectively,  the  images 


x  = 


=  0, 


(54) 


?r 


» 

o          o  o 

in  the  zy-plane.     These  curves  are  shown  in  the  accompanying 


FIG.  9. 


figures,  the  a>axis  being  drawn  triply  between  x  =  0  and  x=  32/3 
since  this  segment  is  described  three  times  by  the  point  (54) 
with  varying  u.  To  the  auxiliary  arc  —  °o<w^—  2,  0=0 
there  corresponds  the  curve 


-2       3-*  ^       -- 

shown  dotted  in  the  figure. 

Consider  now,  for  example,  the  region  a  in  the  wr-plane. 


84  THE  PRINCETON   COLLOQUIUM. 

Its  boundary  is  transformed  into  the  boundary  of  the  region  a 
in  Fig.  10.     According  to  the  generalization  of  the  theorem  of 


FIG.  10. 

Schoenflies  in  §  8,  the  transformation  defines  a  one-to-one 
correspondence  between  the  regions  a  and  a;  and  the  inverse 
functions  u(x,  y),  v(x,  y)  so  defined  are  continuous  over  a.  and 
analytic  in  its  interior. 

Consider  now  the  region  of  points  (u,  v,  x,  y)  defined  by  the 
conditions  that  (u,  v)  shall  lie  in  the  region  6  or  on  its  boundary, 
while  (x,  y)  is  unrestricted.  There  is  but  one  sheet  of  solutions  of 
equations  (53)  in  this  region,  since  any  two  particular  solutions 
(uf,  v',  xe,  y'),  (u",  v",  x",  y")  interior  to  the  sheet  can  be  joined 
by  a  continuous  curve  lying  entirely  within  the  sheet,  as  may  be 
seen  by  joining  (u',  »'),  (u",  v")  by  a  continuous  curve  in  6. 
No  one  of  the  solutions  in  question  has  a  projection  (x,  y)  outside 
of  j3,  since  otherwise  every  point  exterior  to  /3  would  be  such  a 
projection,  according  to  the  third  theorem  of  §  5  or  the  fourth  of 
§  8;  and  from  the  second  of  equations  (53)  it  is  evident  that  no 
solution  (u,  v,  x,  y)  has  a  negative  value  for  y.  On  the  other 


FUNDAMENTAL  EXISTENCE  THEOREMS.  85 

hand  every  point  of  /3  is  the  projection  of  a  solution.  Since  0  is 
simply  connected,  it  follows  from  the  fourth  theorem  of  §  8  that 
the  sheet  of  solutions  is  single-valued  and  that  the  equations  (53) 
define  a  one-to-one  correspondence  between  6  and  /3  similar  to 
that  for  a  and  a. 

A  similar  argument  can  be  made  for  each  of  the  regions  shown 
in  the  figure  and  its  corresponding  image  in  the  xy-plane. 


CHAPTER  III 

EXISTENCE  THEOREMS   FOR  DIFFERENTIAL  EQUATIONS 

It  is  not  within  the  limited  scope  of  these  lectures  to  give  a 
complete  account  of  the  various  methods  for  proving  the  existence 
of  a  system  of  solutions  of  a  set  of  ordinary  differential  equations, 
nor  would  it  be  advisable,  in  view  of  the  many  able  presentations 
of  these  fundamental  theorems  already  well  known  in  mathe- 
matical literature.  It  is  rather  the  intention  of  the  writer  to 
insist  on  conclusions  which  can  be  derived  from  known  methods 
with  regard  to  the  behavior  of  solutions  in  any  region  of  size 
and  shape  compatible  with  the  continuity  properties  of  the 
functions  by  means  of  which  the  equations  are  defined,  as  over 
against  the  usual  restriction  of  the  problem  to  a  rectangular  or 
•circular  neighborhood  of  a  particular  point.  It  has  been  remarked 
by  Picard*  and  Painlevef  that  if  a  continuous  solution  of  the 
differential  equation 


exists  over  an  interval  a  ^  x  ^  /3,  then  the  Cauchy  polygons  of 
approximation  are  defined  and  converge  uniformly  to  the  solution 
for  all  values  of  x  in  the  interval.  In  §  17  below  it  is  shown  that 
in  a  region  R  in  which  the  function  /  is  continuous  and  satisfies 
the  so-called  Lipschitz  condition,  the  polygons  of  Cauchy  pass- 
ing through  a  given  initial  point  (£,  77)  interior  to  R  define  a 
priori  a  continuous  solution  of  the  differential  equation  extending 
to  infinity  or  else  to  the  boundary  of  the  region.  It  follows  then 
that  there  is  a  function 
(2)  _  y  =  <p(x,  £,  77) 

*  Comptes  Itendus,  vol.  128  (1899),  page  1363. 

t  Bulletin  de  la  Socitte  Mathematique  de  France,  vol.  27  (1899),  p.  151. 

86 


FUNDAMENTAL  EXISTENCE  THEOREMS.  87 

satisfying  the  differential  equation  (1)  and  defined  over  a  region 
of  points  (x,  £,  T;)  of  the  form 

(£,  77)  interior  to  R,    a(£,  77)  <  x  <  /3(£,  77), 

and  as  x  approaches  a  or  /3  the  only  limiting  points  which  the 
points  (x,  y)  defined  by  the  function  (2)  can  have  are  at  infinity 
or  else  on  the  boundary  of  the  region  R. 

In  §  18  attention  is  called  to  the  theorems  of  Bendixon  by 
means  of  which  it  can  be  shown  that  the  function  <p  is  continuous, 
and  in  certain  circumstances  differentiable  with  respect  to  the 
arguments  £,  77  as  well  as  with  respect  to  x.  The  "  imbedding 
theorem  "  of  Bolza*  which  asserts  that  any  given  solution,  near 
which  the  function  /  has  suitable  continuity  properties,  can  be 
imbedded  in  a  one-parameter  family  of  neighboring  solutions 
of  the  differential  equation,  is  an  immediate  consequence  of  these 
results,  an  analogue  for  differential  equations  of  the  fundamental 
theorem  for  implicit  functions  proved  in  §  1. 

The  methods  mentioned  above  are  applicable  almost  without 
change  of  wording  to  a  system  of  equations 

~f7A  =  /e  (x,  ?/i,  2/2,  '  *  *>  2/n)       03  =  1,  2,  •  •  -,  ri) 

when  the  symbols  y  and/ in  equations  (1)  are  interpreted  as  row 
letters  in  the  way  apparently  first  introduced  for  differential 
equations  by  Peano.f 

An  interesting  deduction  from  the  theorems  for  a  system  of 
equations  is  the  proof  of  the  existence  of  a  solution  of  a  partial 
differential  equation 

dz     dz 


which  is  not  necessarily  analytic  in  its  five  arguments,  by  means 
of  the  well-known  theory  of  characteristic  curves,  as  described 
in  §  19. 

*  Vorlesungen  iiber  Variationsrechnung,  page  179. 

t  "  Integration  par  series  des  equations  differentielles  line'aires,"  Mathe- 
matische  Annalen,  vol.  32  (1888),  p.  450. 


88  THE  PRINCETON  COLLOQUIUM. 

§  15.    THE  CONVERGENCE  INEQUALITY 
There  is  an  inequality  which  is  of  frequent  service  in  the 

existence  proof  of  the  following  sections  and  which  can  be  readily 

deduced  from  a  simple  preliminary  theorem. 

If  u  is  a  single-valued  function  of  i  with  a  well-defined  forward 

derivative  u'  at  each  point  of  the  interval  0  ^  t  ^  t\,  and  if 

\u'\  <  k\u  +  /, 

k  and  I  being  two  positive  constants,  then  u  also  satisfies  the 
inequality 

M  ^  \u0\ekt  +  ^  (ekt  -  1), 

where  u0  is  the  initial  value  of  u  at  t  =  0. 
Consider  the  function 

v  =  \u0\ekt  +  ~  (ekt  -  1) 

satisfying  the  differential  equation 

v'  =  kv  +  I 

and  having  u0\  as  its  initial  value.  The  value  of  u  is  never 
greater  than  that  of  v,  since  otherwise  the  difference  u  —  v 
would  vanish  and  have  a  positive  or  vanishing  forward  derivative 
at  some  point.  At  a  point  where  u  and  v  are  equal,  however, 

M'|  <  k\u  +  I  =  kv  +  /  =  v', 

which  is  a  contradiction.  A  similar  argument  shows  that  —  u 
is  always  less  than  v. 

If  u  is  a  single-valued  function  of  x  with  well-defined  forward  and 
backward  derivatives  at  each  point  of  an  interval  x0  ^  x  ^  x\, 
and  such  that 


u 


k\u 


then,  for  any  £  and  x  in  the  interval,  u  also  satisfies  the  inequality 
(3)  u\  ^  |tt($)|e*"-«   +  -I  («*'-*'  ~  1). 

K 


FUNDAMENTAL   EXISTENCE   THEOREMS.  89 

This  may  be  proved  from  the  preceding  paragraphs  by  putting 
t  =  x  —  %  for  values  of  x  greater  than  £,  and  t  =  —  x  -f-  £ 
for  values  less  than  £. 

§  16.    THE  CAUCHY  POLYGONS  AND  THEIR  CONVERGENCE  OVER 
A  LIMITED  INTERVAL 

It  is  proposed  to  consider  a  differential  equation  (1)  for  which 
the  function  f(x,  y)  is  continuous  in  the  interior  of  a  certain  region 
R  of  the  xy-p\ane,  and  such  that  the  quotient 

f(x,yT)-f(x,y) 

y'-y 

is  finite  when  (x,  y)  and  (x,  y'}  lie  in  any  closed  region  whose 
points  are  all  interior  to  R. 

A  so-called  Cauchy  polygon  for  the  equation  (1)  through  a 
point  (£,  77)  interior  to  R  is  defined  by  means  of  equations  of  the 
form 

y\  = 


y  =  yn-i+f(xn-i,  yn-\)(x  —  zn_i). 

The  division  points 

£  <  xi  <  x2  < 

may  be  taken  for  convenience  at  equal  distances  5  from  each 
other.  Any  value  x  >  £  will  lie  on  one  of  the  intervals  xn-ixn, 
and  the  polygon  will  either  be  well-defined  for  all  such  values, 
or  else  there  will  be  a  constant  ft  such  that  for  every  x  in  the  in- 
terval £  ^  x  <  ft  the  points  of  the  polygon  are  interior  to  R, 
while  for  x  =  ft  the  corresponding  point  (x,  y)  will  be  a  point  of 
the  boundary  of  R.  The  polygon  defined  by  the  equations  above 
may  be  denoted  by  PI(X),  and  the  analogous  one  when  the  division 
points  are  distant  5/2n-1  from  each  other  by  Pn(x). 

A  common  interval  £  ^  x  ^  a  for  two  functions  P(x),  Q(x) 
with  respect  to  any  region  R  may  be  defined  as  one  over  which 


90  THE   PRINCETON   COLLOQUIUM. 

both  are  interior  to  R,  and  one  such  that  on  any  ordinate  of  the 
interval  all  the  points  between  (x,  P(a*))  and  (x,  Q(x))  are  also 
interior  points  of  R. 

Consider  now  a  closed  region  RI  interior  to  R  and  containing 
the  point  (£,  77),  and  let  m  and  k  be  two  constants  greater  respec- 
tively than  the  absolute  values  of  f(x,  y)  and  the  quotient  (4) 
in  the  region  R\.  If  I  >  0  is  given  in  advance,  the  partitions  for 
any  tico  polygons  P(x},  Q(x)  through  (£,  77)  can  be  taken  so  small 
that 

(5)  \P(x)-Q(x)\  £  jfr*-*1--  1) 

for  all  values  of  x  in  any  common  interval  of  P(x)  and  Q(x)  with 
respect  to  RI.  For  at  the  point  (x,  y),  where  y  =  P(x),  the  equa- 
tion 

P'  =  f(x,  P)  +  {/(*„_!,  2/n_0  -  f(x,  P)}  =  f(x,  P)  +  p 

is  satisfied  by  the  forward  and  backward  derivatives  of  the 
polygon  P.  On  account  of  the  continuity  of  f(x,  y)  there  exists 
for  any  /  a  constant  /*  such  that 


x—  x 
imply 


\y—y'< 


whenever  the  points  (x,  y)  and  (x,  y')  are  in  RI.  If  the  subdivi- 
sions for  P(z)  are  taken  less  than  /z  and  nfm  in  length,  it  follows 
that  on  the  polygon  P(x) 


x  — 


—  yn-\ 


m 


x  —  xn- 


and  hence  the  absolute  value  of  p  is  less  than  1/2.     Similarly 
Q(x)  satisfies  an  equation 


where    a  <  1/2,  provided  that  its  intervals  are  less  in  length 
than  ju  and  n/m.     The  difference  P  —  Q  has  forward  and  back- 


FUNDAMENTAL   EXISTENCE  THEOREMS.  91 

ward  derivatives  which  satisfy  the  relations 

\P'-Q'\  £|/(*,P)-/(*,<?)|  +  |P|  +  *• 
<  k\P  -Q\  +  l, 

and  with  the  help  of  the  lemma  of  §  15  the  desired  inequality 
follows  at  once,  since  P  and  Q  have  the  same  initial  value  rj  at 

*-*. 

If  P(x)  is  a  polygon  and  Q(x)  a  solution  of  the  differential 
equation,  or  if  both  are  solutions,  the  same  theorem  evidently 
holds  true,  because  then  the  function  a  is  identically  zero,  or  else 
both  p  and  a  vanish. 

The  polygons  Pn(x)  all  have  a  common  interval.  For  take 
positive  constants  a  and  b  such  that  the  rectangle 

(6)  0  ^  x  -  £  <;  a,         \y  -  rj    ^  b 

is  entirely  within  R,  and  consequently  has  two  constants  m  and 
k  analogous  to  those  above  for  RI.  The  portions  of  the  polygons 
in  the  rectangle  (6)  all  lie  between  the  straight  lines 

y  —  77  =  =*=  m(x  —  £), 

since  the  slope  of  any  side  of  any  one  of  them  is  numerically 
less  than  m.  It  follows  that  each  is  certainly  well  defined  and 
within  the  rectangle  over  an  interval  £  ^  x  ^  ai,  where  ai  is 
the  smaller  of  a  and  b/m. 

The  sequence  of  polynomials  Pn(x)  converges  uniformly,  on  the 
interval  £  5±  x  ^  £  +  ai,  to  a  function  y(x)  which  has  a  continuous 
derivative  and  satisfies  the  differential  equation  (1).  The  curve 
y  =  y(x)  so  defined  is  entirely  within  the  region  R. 

For  take  e  >  0  arbitrarily,  and  /  so  small  that 


Then 


i_  1    <  €. 


|Pn,(*)  -  P«Gr)|  <      {«*•'  -  1}  <  6, 


92  THE   PRINCETON   COLLOQUIUM. 

provided  that  the  intervals  6/2n' -1  and  5/2""1  are  each  less  than 
the  constant  /x  corresponding  to  /.  Hence  the  sequence  Pn(x) 
converges  uniformly  to  a  continuous  function  y(x]  on  the  interval 

The  equations 

Pn(x)    =    77  +     f  Pn'Wdx    =    77  +     f    {/(*,  Pn)  +  Pn}dx 
J(  Jt 

hold  for  every  n,  and  the  sequences  {f(x,  Pn) }  and  { pn }  approach 
uniformly  the  limits  f(x,  y(x})  and  zero,  respectively.  Hence 


y(x)  =  77  +  J  /(a:, 


from  which  it  follows  by  differentiation  that  y(x)  is  a  solution  of 
the  differential  equation. 

It  is  easy  to  show  by  means  of  the  convergence  inequality 
that  there  is  only  one  continuous  solution  y  =  y(x)  of  the  dif- 
ferential equation  (1)  in  the  region  R  and  passing  through  (£,  77). 
For  suppose  there  were  another,  Y(x),  distinct  from  y(x)  at  a 
value  x'  >  £.  There  would  then  be  a  value  £1  <  x'  at  which 
2/(£i)  =  Y(l-i),  and  such  that  the  two  solutions  would  be  distinct 
throughout  the  interval  £1  <  x  ^  x'.  In  a  neighborhood  of 
the  point  of  intersection  (£1,  771)  interior  to  R  a  relation 

H/(*,  Y)  -  f(x,  y)\  <  k\Y  -  y\ 


dx 

would  be  satisfied,  and  hence,  from  the  convergence  inequality  (3), 

Y  -  y\  ^  0. 

This  contradicts  the  hypothesis  that  y(x)  and  Y(x)  are  distinct 
throughout  the  interval  £1  <  x  ^  x'. 


FUNDAMENTAL  EXISTENCE  THEOREMS.  93 

§  17.    THE  EXISTENCE  OF  A  SOLUTION  EXTENDING  TO  THE 
BOUNDARY  OF  THE  REGION  R 

It  has  been  proved  in  the  preceding  section  that,  on  a  certain 
interval  £  ^  x  ^  £  +  a\,  the  polygonal  curves  y  =  Pn(x)  con- 
verge uniformly  to  a  continuous  solution  y  =  y(x)  of  the  differ- 
ential equation  (1)  lying  entirely  within  the  region  R.  The  in- 
terval for  which  the  proof  has  been  given  may  not  be  the 
largest  one  on  which  the  sequence  of  polygons  has  this  property. 
There  will,  however,  be  a  number  ft  ^  £  +  a\,  possibly  infinity, 
with  the  property  that  on  any  interval  £  ^  x  ^  fti,  where 
ft\  <  ft,  the  sequence  of  polygons  converges  uniformly  to  a 
continuous  solution  interior  to  R.  A  continuous  curve  y=y(x) 
is  thus  defined  which  has  a  derivative  and  satisfies  the  differential 
equation  for  all  values  of  x  in  the  interval  £  ^  x  <  ft. 

As  x  approaches  ft  the  points  (x,  y(x}}  of  the  solution  can  have 
no  limit  point  (ft,  7)  interior  to  the  region  R. 

If  they  did,  there  would  be  for  any  given  c  a  value  x'  <  ft 
such  that 


\y(x'}  -  y 


, 


and  an  integer  N  such  that,  whenever  n  ^  N,  the  inequality 


would  hold  for  all  values  of  x  in  the  interval  £  ^  x  ^  x'.  At  the 
value  x'  in  particular 

|Pn(*')  -  7    ^  |P»GO  -  y(x')\  +  \y(x')  -  7   <  e; 

so  that  for  n  ^.  N  the  points  (#',  Pn(x'}}  would  all  lie  in  the 
e-neighborhood  of  the  point  (ft,  7).  About  the  point  (ft,  7)  as 
center  a  rectangle 

x  -  ft\  ^  A,         \y-y\£B 
could  be  described  entirely  within  the  region  R,  and  in  the  portion 


94  THE   PRINCETON   COLLOQUIUM. 

RI  of  R  which  lay  within  the  rectangle  or  within  the  region 

SZxZx',         2/(.r)  -  e  ^  y  ^  y(x)  +  e 

the  absolute  values  of  f(x,  y)  and  the  quotient  (4)  would  be  less 
than  two  constants  m  and  k,  respectively.  It  can  be  shown 
without  great  difficulty  that  every  polygon  Pn(x)  for  n  ^  N 
would  be  defined  and  lie  within  the  region  R  for  an  interval 
extending  beyond  ft  at  least  a  distance  A\,  where  A\  is  the  smaller 
of  the  numbers  A  and  (B  —  e  —  me)/w.  A  proof  similar  to  that  of 
§  16  would  then  show  that  the  polygons  Pn(x)  converge  uniformly 
to  a  continuous  solution  of  equation  (1)  interior  to  R\  over  an 
interval  £  £  x  £  ft -{•  Ail  and  consequently  ft  could  not  be  the 
upper  bound  described  above. 

As  x  approaches  ft,  therefore,  the  only  limiting  points  of  the 
solution  y  =  y(x)  are  at  infinity  or  else  are  boundary  points  of 
the  region  R.  If  R  is  further  a  closed  region,  that  is,  one  con- 
taining all  of  its  limit  points,  then  there  is  but  one  limit  point 
for  the  curve  y  =  y(x)  as  x  approaches  ft.  For  suppose  (ft,  7) 
to  be  a  finite  point  in  any  neighborhood  of  which  there  are  points 
on  the  curve.  About  (ft,  7)  a  rectangle 

(7)  \x-ft\^A,        \y  -  7   ^  B 

can  be  chosen  arbitrarily,  and  the  points  of  R  lying  in  it  form  a 
finite  closed  set  in  which  \f(x,  y)\  remains  always  less  than  a 
constant  M.  On  the  interval  ft  —  A\  <  x  <  ft,  where  A\  is 
the  smaller  of  the  numbers  A  and  B/M,  all  the  points  of  the 
curve  y  =  y(x)  satisfy  the  inequality 

(8)  \y  -  7!  £  M(ft  -  x}. 

For  if  (xf,  y')  is  any  point  of  the  curve  in  the  rectangle  (7)  and 
also  in  an  e-neighborhood  of  the  point  (ft,  7),  then  the  inequality 

\y  -  y\  ^  \y'  -  2/1  +  \y'  -  T| 

<  M(x'  -  x)  +  e 


FUNDAMENTAL   EXISTENCE   THEOREMS.  95 

must  be  satisfied  by  any  preceding  point  P  (x,  y)  of  the  curve 
y  =  y(x)  for  which  the  arc  PPf  is  interior  to  the  rectangle.  It 
follows  that  the  solution  must  lie  interior  to  the  rectangle  and 
satisfy  the  last  inequality,  at  least  on  an  interval  x'—  A,<x<x', 
where  At  is  the  smaller  of  A  —  e  and  (B  —  €)(M.  Hence  the 
inequality  (8)  is  also  true  on  a  properly  chosen  interval  preceding 
x  =  j8.  It  follows  that  as  x  approaches  /3  there  can  be  but  one 
limit  point  for  the  curve  y  =  y(x),  and  this  limit  point  is  either  at 
infinity  or  else  is  a  boundary  point  of  the  region  R 
When  the  function  f(x,  y)  in  the  differential  equation 


-, 

satisfies  in  a  region  R  the  conditions  stated  at  the  beginning  of  §  16, 
there  exists  through  any  interior  point  (£,  77)  of  the  region  R  one 
and  but  one  continuous  solution 

(9)  y  =  <P(X,  £,  T?) 

of  the  differential  equation.  This  solution  is  defined  and  interior 
to  R  for  all  values  of  x  interior  to  an  interval 

(10)  «(£,  77)<  x  <  |8(€,  17), 

while  as  x  approaches  one  of  the  end  values  a  or  /3,  the  only  limiting 
points  of  the  solution  are  either  at  infinity  or  else  on  the  boundary  of 
R.  If  the  region  R  is  closed,  then  the  solution  has  a  unique  finite  or 
infinite  limit  point  as  x  approaches  a  or  /3. 

§  18.    THE  CONTINUITY  AND  DIFFERENTIABILITY  OF  THE 
SOLUTIONS 

It  can  be  shown  by  methods  due  to  Bendixon*  that  the  func- 
tion <p(x,  £,  TJ)  and  its  derivative  <px(x,  £,  77),  whose  existence  has 
been  proved  in  the  preceding  sections,  are  continuous  in  all  three 
of  their  arguments,  and  if  the  function  f(x,  y)  has  continuous  first 
derivatives  with  respect  to  x  and  y  in  the  interior  of  the  region  R, 

*  Bulletin  de  la  Socitte  MatMmatique  de  France,  vol.  24  (1896),  p.  220. 


96 


THE   PRINCETON   COLLOQUIUM. 


then  <p  and  <px  have  also  continuous  first  derivatives  with  respect 
to  all  of  their  arguments. 

The  continuity  at  any  set  of  values  (x,  £,  77)  for  which  (£,  77) 
is  in  R  and  x  satisfies  the  inequality  (10)  is  provable  with  the 
help  of  the  convergence  inequality  of  §  15.  For  there  will 
always  be  a  region  Rs  about  the  arc  S  of  the  solution  (9)  over  the 


FIG.  11. 


interval  from  £  to  x,  of  the  kind  symbolized  in  the  figure,  and  so 
small  that  it  lies  entirely  within  the  region  R.  If  (£+A£,  77+  AT?) 
is  any  point  in  R,  then  the  solution 


OS) 


y  =  <P(X,  £  +  A£,  77  +  AT?) 


satisfies  the  inequality 
(11) 


AT?)  - 


^  |AT?|  +  m|A$|, 

where  m  is  the  maximum  of  the  absolute  value  of  f(x,  y}  in  Rs, 
on  account  of  the  relation 


(12)     li- 


I 


Hence  as  long  as  S  remains  within  the  region  Rs,  it  satisfies  the 


FUNDAMENTAL   EXISTENCE   THEOREMS.  97 

convergence  inequality 

\<p(x,  S  +  A£,  T;  +  AT;)  -  V(x,  €,  >?)| 


the  initial  values  of  the  two  solutions  being  taken  at  x  =  %  +  A£. 
If  A£  and  AT;  are  sufficiently  small  the  expression  on  the  right 
is  less  than  5  for  all  values  of  x  belonging  to  the  region  Rs,  and 
hence  S  must  be  defined  and  interior  to  R&  for  all  such  values. 
Otherwise,  for  some  interior  value  of  x,  it  would  attain  one  of  the 
values  (f>(x,  £,  77)  =*=  8,  which  is  seen  to  be  impossible  on  account 
of  the  choice  just  made  of  A£  and  AT;. 
Consider  now  the  difference 


• 

By  a  step  similar  to  (12),  and  the  inequality  (11),  it  is  seen  to  be 
less  than 

{|ATT|  + 


whenever  A£  and  AT;  have  been  so  chosen  that  S  lies  entirely  in 
the  region  Rs.     Hence  the  continuity  of  <p(x,  £,  77)  is  proved. 

To  prove  the  differentiability  of  <p  with  respect  to  £  and  TT, 
assume  that  f(x,  y)  has  a  continuous  derivative  fv  in  the  region 
R,  and  consider  the  same  solutions  S  and  /S  in  the  region  Rs. 
The  difference  of  their  ordinates  satisfies  the  equation 


=  f(x,  v  +  W-  f(x,  &  = 


where,  by  Taylor's  formula  with  the  integral  form  of  remainder, 

A  =    I    fy(x,  <f>  + 
Jo 


is  a  continuous  function  of  x,  A£,  AT;,  the  values  £,  rj  being  con- 

8 


98  THE  PRINCETON   COLLOQUIUM. 

sidered  as  constant  for  the  moment.     Hence 

A^  =  ceS****. 

When  A  £  =  0  or  AT;  =  0,  the  constant  c  has  respectively  the  values 
(t,  £,  77  +  AT;)  —  <p(£,  £,  17)  =  A 
,  T;)  -  <p(£,  £,  T;)  =    I        f(x,  <f> 

i/        A 


c  =  &<p\x=t  =  <p(t,  £,  77  +  AT;)  —  <p(£,  £,  17)  =  AT;, 


where  0  <  6  <  1.     Hence  the  quotients  A<p/A£,  A<P/AT;  have  well- 
defined  limiting  values 

d(f> 


It  may  be  remarked  in  conclusion  that  the  theorems  which 
have  been  proved  in  §§  16-18  are  true  for  systems  of  equations 
as  well  as  for  a  single  one. 

§  19.    AN  EXISTENCE  THEOREM  FOR  A  PARTIAL  DIFFERENTIAL 

EQUATION  OF  THE  FIRST  ORDER  WHICH  is 

NOT  NECESSARILY  ANALYTIC 

Proofs  have  been  given  by  Cauchy,  Kowalewski,  Darboux, 
and  others  for  -the  theorem  that  in  general  there  exists  one  and 
but  one  analytic  surface 

2  =  z(x,  y} 

which  passes  through  an  arbitrarily  selected  analytic  curve  C 
in  the  ary-space  and,  with  the  derivatives 

_  dz  dz 

P  =  dx'        q=dy> 

satisfies  a  differential  equation  of  the  form 
F(x,  y,  z,  p,  9)  =  0, 


FUNDAMENTAL  EXISTENCE  THEOREMS.  99 

where  F  is  an  analytic  function  of  its  five  arguments.  These 
proofs,  however,  say  nothing  about  the  solutions  which  may 
exist  through  a  curve  C  whose  defining  functions  are  not  ex- 
pressible by  means  of  power  series;  and  they  are  not  applicable 
when  F  itself  has  not  this  property.  An  existence  proof  is  to 
be  given  below  which  is  based  upon  much  less  restrictive  as- 
sumptions on  the  functions  F  and  the  curve  C.  It  involves  the 
well-known  theory  of  characteristic  strips,  which  are  solutions 
of  a  set  of  ordinary  differential  equations.  If  a  one-parameter 
family  of  characteristic  strips  intersecting  a  given  curve  C  is 
properly  selected,  it  will  generate  a  surface  S  which  is  a  solution 
of  the  differential  equation.  The  existence  of  the  family  and  the 
differentiability  of  the  surface  depend,  however,  upon  the 
existence  and  differentiability  of  the  equations  of  the  character- 
istic strips  with  respect  to  the  initial  values  of  the  variables 
which  they  involve,  that  is,  upon  theorems  similar  to  those  which 
have  been  developed  in  the  preceding  sections. 

Suppose  that  the  function  F  is  continuous  and  has  continuous 
first  and  second  derivatives  in  a  certain  region  R  of  points 
(x,  y,  z,  p,  0.  The  differential  equations  satisfied  by  the  charac- 
teristic strips  have  the  form 

dx  dy  dz 

du  =  F"    <lu  =  F«'    du  =  PF°  +  1F« 
(13) 

%--'.-*•  J--A-*, 

Through  any  initial  values  (£,  77,  f,  TT,  K)  interior  to  R  these 
equations  have  a  solution  with  equations  and  initial  conditions 
of  the  form 

x  =  x(u,  £,  77,  f,  TT,  K),         £  =  x(Q,  £,  77,  f,  TT,  K), 

y  =  y(u,  £,  -n,  r,  *,  *),       -n  =  y(Q,  £,  77,  f,  x,  *), 

(14)         2  =    Z(U,   £,  77,  f,  7T,   K),  f   =    2(0,   £,  77,  f,  7T,  K), 

P  =  P(U,  £,  r?i  T,  if,  K),         TT  =  p(0,  £,  77,  f,  TT,  K), 

<?  =    ?(W,    £,  77,   f,   7T,    K),  K  =    ^(O,    £,  77,   f,  X,    K), 


100  THE    PRINCETON   COLLOQUIUM. 

and  such  that  each  of  the  functions  on  the  left  and  its  derivative 
for  w  are  continuous  and  have  continuous  first  derivatives  in  a 
region  of  values  (?/,  £,  77,  f,  TT,  K)  for  which  (£,  77,  f,  TT,  K)  is  a 
point  interior  to  R  and  ?/  lies  in  an  interval,  containing  the  value 
u  =  0,  of  the  form, 

«(£,   -n,   f,  T,    O<  W  <  0(£,  77,   f,  7T,    K). 

The  points  (ar,  y,  2,  p,  7)  so  defined  are  all  interior  to  the  region  R. 
Along  the  solution  (14)  the  equations 

(15)  pxu  +  qyu  —  zu  =  0 

dF 

(16)  =  Fzxu  +  />„  +  Fz2u  +  *>u  +  />„  =  0 


are  satisfied  identically,  so  that  the  direction  p  :  q  :  —  1  is  always 
normal  to  the  curve  defined  by  the  first  three  equations.  Evi- 
dently if  F  vanishes  at  a  single  point  of  the  strip,  it  will  also 
vanish  at  every  other  point.  The  solutions  (14)  along  which  F 
vanishes  are  called  characteristic  strips,  and  any  one  of  the 
strips  (14)  will  surely  be  of  this  type  if  the  initial  condition 


7,  r,  T,  K)  =  o 

is  satisfied. 

Consider  now  a  continuous  and  differentiable  strip  of  elements 


(17)     x  =  £(r),     y  =  rj(v),     z  =  f(r),     p  =  7r(r),     9  =  /c(r) 

which  lies  in  the  interior  of  the  region  R  and  satisfies  the  con- 
ditions 

v     i  >•   ._  A 

TT^t!  ~r  kTjt,  —  5r  —    ' 
(18) 


where  the  arguments  in  the  derivatives  of  F  are  the  same  as  those 
in  the  second  equation.  The  first  two  of  these  conditions  imply 
that  the  direction  TT  :  K  :  —  1  is  normal  to  the  curve 

(19)  x  =  £00,    y  =  i?(t),    2  =  f (»), 


FUNDAMENTAL  EXISTENCE  THEOREMS.  101 

and  that  the  curve  and  its  strip  of  normals  satisfy  the  differential 
equation.  The  third  prevents  the  strip  from  being  a  so-called 
integral  strip  of  the  differential  equation,  through  which  there 
does  not  in  general  pass  a  unique  integral  surface  without 
singularities.  To  make  the  situation  simpler  it  will  be  supposed 
that  the  projection  of  the  strip  (17)  in  the  xy-p\ane  does  not 
intersect  itself. 

When  the  functions  (17)  are  substituted  in  the  equations  (14), 
a  new  system 

x  =  X(it,  v),    y  =  Y(u,  v),    z  =  Z(u,  v), 
p  =  P(u,  v),     q  =  Q(u,  v) 

with  the  initial  conditions 

»,     77(0)  =  F(0,  »),     f(tO  =  Z(0,  t>), 


T(»)  =  P(0,  v),     K(V)  = 
is  determined.     There  is  a  region 

(/?«„)  A    ^U^B,       »!    ^   V   £   Vz, 

where  A  is  a  negative  and  B  a  positive  constant,  in  which  the 
functions  (20)  are  continuous,  have  continuous  first  derivatives, 
and  satisfy  the  relation 


(22) 


4=  0. 

y     y 

•*  u        -f  v 


For  if  M  is  the  maximum  of  the  absolute  values  of  the  functions 
on  the  right  in  the  equations  (13),  for  a  closed  c-neighborhood  of 
the  points  of  the  strip  (17)  in  the  interior  of  R,  then  the  solutions 
(14)  are  defined  at  least  over  an  interval  u  ^  c/3/,  and  the 
absolute  values  of  A  and  B  can  be  taken  at  least  as  great  as  this 
constant  without  disturbing  the  continuity  properties  desired 
for  the  functions  (20)  in  the  region  Ruv.  The  condition  (22)  is 
satisfied  for  the  values  u  =  0,  t'i  ^  v  ^  ^  because  of  the  first 
two  of  equations  (13)  and  the  third  of  the  relations  (18);  and  the 


102  THE   PRINCETON   COLLOQUIUM. 

region  Rvv  can  therefore  be  chosen  so  that  the  determinant  is 
different  from  zero  everywhere  in  it. 

By  an  argument  similar  to  that  used  in  proving  the  theorem 
of  §  4  it  can  be  shown  that  A  and  B  can  be  restricted  still  further, 
if  necessary,  so  that  no  two  distinct  points  (ur,  v'),  (u",  v")  in 
the  region  Ruv  define  the  same  point  (.r,  y)  by  means 
of  equations  (20).  The  boundary  of  the  region  Ruv  is  trans- 
formed then  by  the  first  two  of  equations  (20)  into  a  simply 
closed  regular  curve  in  the  zy-plane  which  bounds  a  portion 
Rxy  of  the  a*t/-plane.  The  equations  establish  furthermore  a 
one-to-one  correspondence  between  the  points  of  Ruv  and  those 
of  Rxy,  and  the  functions 

(23)  n  =  u(x,  y),    v  =  v(x,  y) 

so  defined  are  continuous  and  have  continuous  first  derivatives 
in  Rxy.  The  others  of  the  equations  (20)  define  then  three 
functions 

(24)  2  =  z(x,  y),    p  =  p(x,  y),     q  =  q(x,  y) 

which  are  also  continuous  and  have  continuous  first  derivatives 
in  Rxy,  and  which  with  the  values  (23)  for  u  and  v  satisfy  the 
equations  (20)  identically  in  x,  y. 

The  functions  (20)  satisfy  the  relations 


(25)  PXV  +  QYV  -  Zv  =  0, 

F(X,  7,  Z,  P,  Q)  =  0, 

identically  in  u,  v.  The  first  and  third  of  these  follow  at  once 
from  the  equations  (15),  (16),  the  second  of  the  equations  (18), 
and  (21).  The  expression 


QYV-  Zv 
has  the  initial  values 

(26)  0(0,  v)  =  TT&,  +  w,  -  r,  =  o, 


FUNDAMENTAL   EXISTENCE   THEOREMS. 


103 


which  vanish  on  account  of  the  first  of  equations  (18).     Further- 
more 

12U  =  PUXV  +  QUYV  +  PXUV  +  QYUV, 


and  from  the  first  of  equations  (25), 

0  =  PVXU  +  QVYU  +  PXUV 


QYVV. 


By  subtracting  the  last  expression  from  that  for  12U  and  using 
the  equations  (13)  which  the  functions  (20)  satisfy,  it  follows  that 

_3F 

"  dv' 


\u  —  *uXv  -f~  *fu*v         rvXu         *tv*u  — 


in  which  the  arguments  of  the  derivatives  of  F  are  the  functions 
(20).     Hence  with  the  help  of  the  third  of  equations  (25)  and 

the  initial  values  (26), 

-« 

flu  =  -  12F,,     12  =  12(0,  v)e  *  *  *u'=  0. 

The  single- valued  function  z(x,  y)  defined  above  over  the  region 
Rxy  has  the  derivatives 


Zx  = 


Zu      Yu 

Zv    Yv 


Xu 


=  p(x, 


Zy  = 


**•  u 


•A«    I  u 

Xv    Yv 


=  q(x,  y), 


found  by  substituting  the  functions  (23),  (24)  in  the  equations 
(20),  differentiating  the  resulting  identities,  and  applying  the 
first  two  of  the  relations  (25) .  It  satisfies  the  differential  equation 
F  =  0  on  account  of  the  third  of  the  equations  (25).  Further- 
more 

x,    y,    z(x,  y),    p(x,  y),     q(x,  y) 

reduce  to  £,  77,  £",  TT,  K  at  any  point  of  the  strip  (17),  since  at  such 
a  point  w(£,  77)  =  0  and  the  relations  (21)  are  satisfied. 

It  has  been  proved  therefore  that  there  is  a  single-valued 


104  THE  PRINCETON   COLLOQUIUM. 

function 

(27)  z  =  z(x,  y), 

defined  over  a  region  Rxy  of  the  a-y-plane,  which  is  continuous  and 
has  continuous  first  and  second  derivatives,  contains  the  initial 
strip  (17),  and  satisfies  the  differential  equation  F  =  0. 
There  is  no  other  surface 

(28)  z  =  zi(x,  y} 

defined  over  the  region  Rxy  and  having  these  properties.  If  there 
were  such  a  one,  it  would  have  to  contain  all  of  the  points  of  the 
strips  defined  by  equations  (20).  To  prove  this,  suppose  that 
(xf,  yf,  z'y  p',  <?')  is  an  element  belonging  to  one  of  the  strips  (20) 
for  values  (uf,  0'),  and  also  to  the  surface  (28).  The  equations 

(29)  ^  =  Fp(x,  y,  zi,  pi,  91),     ^  =  Fg(x,  y,  zly  plt  qj, 

where  p\  and  q\  are  the  derivatives  of  z\,  have  a  unique  solution 

(30)  x  =  xi(u)y    y  =  1/1  (w) 

reducing  to  x1,  y'  for  the  initial  value  u  =  u'  and  defined  over  an 
interval  u'  —  «  ^  u  ^  u'  +  €.  The  corresponding  equations 

(31)  x  =  xi(u),    y  =  y\(u),    z  =  z^u), 

p  =  pi(u),     q  =  qi(u), 

found  by  substituting  the  functions  (30)  in  z\,  p\,  q\,  define  a 
characteristic  strip.  For  on  the  surface  (28)  the  equations 


Fv  +  Ftq,  +  Fps,  + 

are  identities  in  x,  y,  where  r\y  s\,  t\  are  the  three  second  derivatives 
of  Zi(x,  y).     As  a  result  of  these  identities  and  the  equations  (29), 


FUNDAMENTAL   EXISTENCE   THEOREMS.  105 


<fei  dx          dy 

du  =  plTu  +  ^Tu 


dq\  _       dx  dy 

where  the  arguments  of  the  derivatives  of  F  are  the  functions 
(31).  The  equations  (29)  and  (32)  show  that  the  strip  (31) 
is  a  characteristic  strip.  Its  initial  element  for  u  =  u'  is 
(xf,  y',  z',  p',  q'},  the  same  as  that  for  the  strip  (20)  corresponding 
to  v  =  TO'.  .Hence  the  two  must  coincide  on  the  interval  u'  —  € 
^  u  ^  u'  -}-  c  on  which  both  are  defined. 

The  initial  element  (21)  of  any  one  of  the  strips  (20)  is  by 
hypothesis  on  the  surface  (28).  According  to  the  last  paragraph 
all  of  the  elements  of  the  strip  in  an  interval  u  ^  e  must  also 
lie  on  the  surface,  and  it  follows  that  there  can  be  no  upper 
bound  except  B  for  the  values  of  u  for  which  this  is  true.  If 
u'  <  B  were  such  a  limiting  value,  the  element  (xf,  yr,  z',  p',  q') 
corresponding  to  u'  on  the  characteristic  strip  wrould  also  belong 
to  the  surface,  on  account  of  the  continuity  of  z\(x,  y)  and  its 
derivatives;  and  the  interval  of  coincidence  would  therefore 
be  necessarily  longer  than  0  ^  u  <  u'. 

For  any  point  (x,  y)  in  the  region  Rxy  there  is  but  one  set  of 
values  (u,  »)  solving  the  first  two  of  equations  (20),  and  the  cor- 
responding value  of  z  from  the  third  equation  belongs  to  both 
of  the  surfaces  (27)  and  (28).  The  two  surfaces  must  therefore 
coincide  throughout. 

Suppose  now  that  an  initial  curve  of  the  form  (19)  is  given 
instead  of  the  initial  strip  (17).  If  to  any  value  VQ  defining  a  point 
(£o>  "no,  To)  of  the  curve  there  corresponds  a  direction  TTO  :  KO  :  —  1 
satisfying  the  relations  (18),  and  such  that  (£0>  f]o,  fo>  TTQ,  KO)  ls 
interior  to  R,  then  there  will  be  a  strip  of  elements  of  the  form 
(17)  along  the  curve  containing  these  initial  values  for  v  =  r0. 
For  the  first  two  equations  (18)  have  the  solution  (v0,  TTO,  KO) 


106  THE   PRINCETON   COLLOQUIUM. 

when  their  first  members  are  regarded  as  functions  of  v,  IT,  K, 
and  on  account  of  the  third  relation  (18)  their  functional  de- 
terminant for  TT,  K  does  not  vanish  at  these  values.  According 
to  the  fundamental  theorem  of  §  1  there  is  therefore  a  pair  of 
functions  TT(V),  K(V)  defined  over  an  interval  v\  ^  n  ^  v?  con- 
taining v0  and  satisfying,  with  £(»),  rj(v),  f(u),  the  relations  (18). 

The  results  of  the  preceding  paragraphs  may  be  summarized 
as  follows: 

Suppose  that 

(C)  x  =  $(«),     y  =  17(0),     2  =  f(t) 

is  a  continuous  and  differentiate  curve,  at  some  point  (£o,  *?o,  To) 
=  (£(*>o),  *?(0o),  r(*o))  of  which  there  is  a  normal  TO  :  KO  :  —  1 
satisfying  the  equation 


r)0,  fo,  TTO,  KO)  =  0. 
Suppose  furthermore  that 


4=0, 


and  that  the  initial  element  (£o>  "no,  To*  TTO>  KO)  lies  in  a  region  R  of 
points  (x,  y,  z,  p,  q)  in  which  F  is  continuous  and  has  continuous 
first  and  second  derivatives.  Then  there  is  a  strip  of  the  form 

(S)      x  =  £(»),     y  =  77(0),     z  =  £(»),     p  =  TT(V),     q  =  K(V) 

vi       v       ^ 


containing  (%0,  y0,  $"0,  TO,  KO)  /or  t?  =  r0,  an</  *z/cA  that  all  of  ite 
elements  have  the  properties  ascribed  above  to  this  initial  one.  If 
the  projection  dy  of  C  in  the  xy-plane  does  not  intersect  itself,  the 
characteristic  strips  of  the  differential  equation 

F(x,  y,  z,  p,q)  =  0 
which  pass  through  the  elements  of  S  simply  cover  a  region  Rxy  of 


FUNDAMENTAL  EXISTENCE  THEOREMS.  107 

ihe  xy-plane  and  envelop  a  single-valued  surface 

z  =  z(x,  y). 

This  surface  is  continuous  and  has  continuous  first  and  second 
derivatives  in  Rxv,  contains  the  strip  S,  and  satisfies  the  differential 
equation  F  =  0.  There  is  no  other  surface  over  the  region  Rzy  which 
has  these  properties. 


DIFFERENTIAL-GEOMETRIC 
ASPECTS  OF  DYNAMICS 


BY 

EDWARD   KASNER 


CONTENTS 


Pages 

INTRODUCTION 1 

CHAPTER  I 

TRAJECTORIES  IN  AN  ARBITRARY  FIELD  OF  FORCE 

1-8.     Trajectories  in  the  plane 7 

9.     Actual  and  virtualt  rajectories 16 

10-15.     Trajectories  in  space 17 

16-25.     The  inverse  problem  of  dynamics:    a  method  of 

geometric  exploration 22 

26-27.     Tests  for  a  conservative  field 31 

CHAPTER  II 

NATURAL  FAMILIES:  THE  GEOMETRY  OF  CONSERVATIVE 
FIELDS  OF  FORCE 

28.     Origin  and  application  of  the  natural  type 34 

29-31.     Characteristic  properties  A   and  B 37 

32.  General  velocity  systems 42 

33.  Reciprocal  systems 44 

34.  Character  of  the  transformation  T 46 

35-44.  The  converse  of  Thomson  and  Tait's  theorem ....  47 
45-53.  Wave  propagation   in  an  isotropic  medium:  pro- 
perties of  wave  sets 57 

54-61.     A  second   converse  problem  connected  with  the 

Thomson-Tait  theorem 61 

62-67.     Geometric  formulation   of    some  curious  optical 

properties 65 

68-72.     The  so-called  general  problem  of  dynamics 69 

i 


11  CONTENTS. 

CHAPTER  III 
TRANSFORMATION  THEORIES  IN  DYNAMICS 

73-81.  Projective  transformations 73 

82-91.  Conformal  transformations 81 

92-94.  Contact  transformations 87 

95-97.  A  group  of  space-time  transformations 89 

CHAPTER  IV 

CONSTRAINED  MOTIONS  IN  A  FIELD.     GENERALIZATION  OF  THE 

TRAJECTORY  PROBLEM  INCLUDING  BRACHISTOCHRONES 

AND  CATENARIES 

98-114.  Systems  Sk  defined  by  P  =  kN 91 

115-116.  Curves  of  constant  pressure 97 

117-118.  Tautochrones 98 

119.  Non-uniform  catenaries 100 

CHAPTER  V 

MORE  COMPLICATED  TYPES  OF  FORCE 

120-122.     Motion  in  a  resisting  medium 102 

123-126.     Particle  on  a  surface 104 

127-130.     The  general  field  in  space  of  n  dimensions 107 

131-132.     Interacting  particles  in  the  plane  and  in  space.  .    109 
133-141.     Forces  depending  on  the  time.     Trajectories  and 

space-time  curves Ill 


DIFFERENTIAL-GEOMETRIC 
ASPECTS  OF  DYNAMICS 


BY 

EDWARD  KASNER 


INTRODUCTION 

The  relations  between  mathematics  and  physics  have  been 
presented  so  frequently  and  so  adequately  in  recent  years,  that 
further  discussion  would  seem  unnecessary.  Mathematics, 
however,  is  too  often  taken  to  be  analysis,  and  the  role  of  geom- 
etry is  neglected.  Geometry  may  be  viewed  either  as  a  branch 
of  pure  mathematics,  or  as  the  simplest  of  the  physical  sciences. 
For  our  discussion  we  choose  the  latter  point  of  view:  geometry 
is  the  science  of  actual  physical  or  intuitive  space.  All  physical 
phenomena  take  place  in  space,  and  hence  necessarily  present 
geometric  aspects.  We  confine  our  discussion  to  mechanics, 
and  consider  the  role  of  geometry  in  mechanics. 

The  fundamental  concepts  of  mechanics  are:  space,  time,  mass, 
and  force.  Certain  preliminary  theories  deal  with  some  instead 
of  all  these  concepts.  Space  by  itself  gives  rise  to  pure  geometry 
with  all  its  subdivisions.  According  to  Sir  William  Rowan 
Hamilton,  algebra  is  the  science  of  pure  time;  in  fact  time  is 
the  simplest  one-dimensional  manifold  suggesting  the  notion 
of  real  number,  the  continuum,  the  foundation  of  analysis. 
Neither  mass  by  itself,  nor  force  by  itself,  gives  rise  to  an  inde- 
pendent theory,  for  these  notions  cannot  be  considered  without 
considering  space  also. 

Space  and  time  together  give  rise  to  kinematics.  If  we  do 
not  consider  velocities  and  accelerations,  but  only  displacements 
(that  is,  initial  and  terminal  positions  without  introducing 
9  1 


2  THE   PRINCETON   COLLOQUIUM. 

continuous  motion  from  one  to  the  other),  we  obtain  Ampere's 
"geometry  of  motion,"  which  belongs  to  pure  geometry  rather 
than  to  kinematics. 

Space  and  mass  give  rise  to  a  separate  discipline  which  may 
be  called  the  geometry  of  masses.  This  deals  with  centers  of 
gravity,  moments  of  inertia,  and  moments  of  higher  type,  which 
have  been  studied  extensively  in  recent  years,  especially  by  the 
Italian  mathematicians. 

Space  and  force  are  the  essential  concepts  employed  in  rigid 
statics.  Mass  and  time  are  not  necessary  in  this  theory,  which 
deals  essentially  with  the  equivalence  and  reduction  of  systems 
of  vectors.  The  remaining  combinations,  mass  and  time,  force 
and  time,  mass  and  force,  do  not  produce  separate  theories, 
since  they  can  not  be  discussed  without  introducing  also  the 
concept  of  space. 

Consider  then  space,  time,  and  mass.  The  principal  develop- 
ment along  this  line  is  Hertz's  remarkable  "geometry  and 
kinematics  of  material  systems,"  a  theory  entirely  independent 
of  the  concept  of  force. 

The  other  combinations  of  three  of  the  four  concepts  have 
not  produced  separate  developments. 

Finally,  we  have  the  theory  which  involves  all  four  concepts 
simultaneously,  namely,  kinetics. 

Although  the  geometric  aspects  of  the  preliminary  theories 
are  very  interesting  and  important,  it  is  not  our  intention  to 
review  the  progress  which  has  been  made  in  this  line.  We 
mention  only  Ball's  theory  of  screws,  Study's  Geometric  der 
Dynamen,  and  the  law  of  duality  connecting  kinematics  and 
statics — a  law  which  is  not  dynamical,  but  purely  geometric. 

The  notion  of  vector  is  of  course  fundamental  in  many  of 
these  theories.  We  recall  the  fact  that  there  are  three  distinct 
types  of  vector  used  in  mechanics:  the  free  vector,  the  sliding 
vector,  the  bound  vector.  These  three  types  differ  with  respect 
to  the  definition  of  equivalence.  In  the  first  theory,  two  vectors 
are  regarded  as  equivalent  when  they  have  the  same  length 


ASPECTS  OF^  DYNAMICS.  3 

and  direction  (including  sense).  Such  free  vectors  are  employed 
in  combining  translations,  or  forces  acting  at  a  point.  A  free 
vector  in  space  has  three  coordinates.  The  sum  of  any  number 
of  vectors  is  a  vector. 

In  the  second  theory,  dealing  with  sliding  vectors,  two  vectors 
are  equivalent  only  when  they  have  the  same  line  as  well  as  the 
same  length  and  sense.  Such  vectors  are  used  in  the  statics  of 
rigid  bodies.  The  sliding  vector  in  space  has  five  coordinates. 
A  system  of  these  vectors  can  not  usually  be  reduced  to  a  single 
vector.  The  most  general  system  depends  in  fact  on  six  essential 
parameters:  it  is  a  new  geometric  element  which  may  be  repre- 
sented either  as  a  screw  or  a  dyname. 

Finally,  in  the  third  type  of  vector  theory,  two  vectors  are 
not  equivalent  unless  they  have  the  same  initial  point  and  same 
terminal  point,  that  is  the  vector  is  completely  bound.  Such  a 
vector  in  space  depends  on  six  coordinates.  The  most  general 
system  depends  on  twelve  essential  parameters.  This  is  the 
theory  required  in  the  developments  of  astatics. 

Statics  and  kinematics  have  given  rise  to  very  extensive 
geometric  developments;  but  kinetics  still  is  thought  of  almost 
exclusively  as  a  matter  of  differential  equations.  Lagrange,  in 
the  famous  preface  to  his  Mecanique  Analytique,  stated  that 
no  diagrams  would  be  found  in  his  wrork:  "Lovers  of  analysis 
will  thank  me  for  adding  a  new  branch  to  that  science."  The 
special  object  of  these  lectures  will  be  to  point  out  some  of  the 
geometric  aspects  of  kinetics,  especially  properties  of  the  tra- 
jectories described  in  arbitrary  fields  of  force.  While  the  in- 
vestigations connected  with  statics  and  kinematics  are  mainly  of 
algebraic-geometric  character,  our  kinetic  discussions  relate  to 
infinitesimal  properties,  tangents,  distribution  of  curvature, 
osculating  conies,  and  so  on:  we  shall  deal  chiefly  with  the 
differential  geometry  of  systems  of  trajectories.  It  is  essential 
to  observe  that  the  properties  considered  relate  not  to  the 
individual  curves,  but  to  the  infinite  systems  of  curves. 

To  emphasize  this  point,  consider  the  motion  of  a  particle 


THE    PRINCETOI^   COLLOQUIUM. 

in  a  plane  field  of  force,  the  force  depending  only  on  the 
position  of  the  point.  For  given  initial  conditions,  the  particle 
will  move  on  a  definite  curve;  taking  all  possible  initial  con- 
ditions, we  shall  obtain  a  triply  infinite  system  of  curves.  A 
single  curve  obviously  has  no  peculiarities,  for  a  particle  may 
be  made  to  describe  any  given  curve  by  selecting  a  proper  force, 
varying  from  point  to  point  of  that  curve.  The  system  of  curves, 
however,  will  have  intrinsic  peculiarities,  for  if  a  triply  infinite 
system  of  curves,is  given  at  random,  it  will  not  usually  be  possible 
to  find  any  field  of  force  such  that  every  particle  moving  in  that 
field  will  describe  one  of  the  given  curves;  there  is,  for  instance, 
no  field  of  force  which  produces  as  its  trajectories  all  the  circles 
of  the  plane. 

The  simplest  general  property  of  the  system  of  trajectories 
is  as  follows:  If  a  particle  is  started  at  a  given  position  in  a  given 
direction  with  all  possible  initial  speeds  into  a  field  of  force,  a 
single  infinity  of  trajectories  will  be  obtained,  one  for  each  value 
of  the  speed;  construct  for  each  of  these  curves  the  parabola 
having  four-point  contact  (osculating  parabola);  the  foci  of 
these  parabolas  will  always  lie  on  a  circle  passing  through  the 
given  initial  point.  An  equivalent  statement  is  that  the  di- 
rectrices of  these  parabolas  will  always  be  concurrent.  In  space 
we  employ  osculating  spheres  and  find  that  the  locus  of  the 
centers  is  a  straight  line. 

A  completely  characteristic  set  of  properties,  for  both  the 
plane  and  space,  is  given  in  Chapter  I.  It  is  thus  possible  to 
tell  when  a  given  system  of  curves  can  serve  as  a  system  of 
dynamical  trajectories.  A  method  is  obtained  for  constructing 
the  field  from  its  trajectories.  If  say  a  handful  of  particles  is 
thrown  into  an  unknown  field  (the  force  acting  at  any  point 
depending  only  on  the  position  of  the  point)  and  if  a  photograph 
of  the  totality  of  paths  is  taken,  then,  without  any  record  of 
velocity  or  any  observation  of  time,  the  field  can  be  constructed. 
In  particular  it  is  possible,  by  simple  geometric  tests,  to  dis- 
tinguish conservative  from  non-conservative  fields. 


ASPECTS    OF   DYNAMICS.  5 

Chapter  II  deals  with  the  geometry  of  conservative  forces. 
Here  the  energy  equation  allows  us  to  group  the  trajectories 
into  "natural  families."  Such  a  family  is  obtained  most  con- 
cretely as  the  totality  of  oo4  rays  or  paths  of  light  in  any  medium 
where  the  index  of  refraction  varies  continuously  from  point  to 
point.  The  geometric  characterization  is  first  given  by  two 
simple  properties  relating  to  circles  of  curvature;  and  then  by  a 
new  converse  of  the  theorem  of  Thomson  and  Tait.  It  is  seen, 
for  example,  that  if  a  candle  is  placed  in  the  atmosphere  or  in 
any  gas  of  variable  density,  the  oo2  rays  emitted  by  it,  which  may 
be  curves  of  very  complicated  shape,  will  necessarily  have  these 
properties:  (A)  the  circles  of  curvature  constructed  at  the  given 
source  all  meet  at  a  second  point;  (B)  three  of  these  circles  have 
four-point  (instead  of  merely  three-point)  contact  with  thsir 
curves,  and  these  thre6  are  mutually  orthogonal;  (C)  the  °o2 
rays  form  a  normal  congruence,  that  is,  admit  oo1  orthogonal 
surfaces.  Natural  families  are  characterized  either  by  (A)  and 
(B),  or  by  (A)  and  (C). 

These  results  are  applied  to  the  propagation  of  waves  in  any 
isotropic  medium.  A  second  and  more  complicated  converse 
question  suggested  by  the  Thomson-Tait  theorem  is  discussed. 
Some  interesting  optical  theorems  are  given  a  geometric  formu- 
lation, but  the  converse  problems  are  left  unsettled.  The  final 
section  deals  with  the  "general  problem  of  dynamics"  in  the 
sense  of  the  French  writers. 

The  third  chapter  deals  with  transformation  theories.  It  is 
interesting  to  notice  how  the  most  important  groups  of  geometry, 
the  protective  and  the  conformal,  play  essential  roles  in  dynamics, 
the  former  in  connection  with  arbitrary  fields,  the  latter  in 
connection  with  conservative  fields  and  natural  families.  The 
infinitesimal  contact  transformations  of  mechanics,  and  a  new 
group  of  space-time  transformations  are  also  discussed. 

The  chief  subject  of  Chapter  IV  is  a  simple  problem  in  con- 
strained motion,  which  includes,  and  hence  serves  to  unify,  the 
theories  of  trajectories,  brachistochrones,  catenaries,  and  velocity 


O  THE    PRINCETON   COLLOQUIUM. 

curves  in  an  arbitrary  field  of  force.  Complete  characteriza- 
tions are  given.  Curves  of  constant  pressure  and  tautochrones 
are  treated  only  briefly. 

Chapter  V  includes  brief  discussions  of  more  complicated 
problems  in  motion,  for  example,  the  effect  of  a  resisting  medium 
on  the  geometric  character  of  the  system  of  trajectories;  the 
motions  of  any  number  of  interacting  particles  (the  results  being 
of  course  applicable  to  the  problem  of  three  bodies);  finally, 
forces  depending  not  only  on  position  but  also  on  the  time,  both 
trajectories  and  space-time  curves  being  studied.  The  latter 
are  constructed,  in  the  sense  of  Minkowski,  in  the  four-di- 
mensional space  (x,  y,  z,  t},  but  the  application  made  is  to  ordinary 
dynamics,  not  to  electrodynamics  or  relativity  theory. 

The  main  results  of  the  first  two  chapters  (in  particular  the 
•complete  characterizations  of  general  systems  of  trajectories  and 
iof  natural  families)  were  first  given  by  the  writer  in  a  series  of 
if  our  papers  published  in  the  Transactions  of  this  Society  (1906- 
1910).  Some  of  the  other  results  are  given  in  notes  published  in 
the  Bulletin.  The  last  two  chapters,  as  wrell  as  many  sections  of 
the  other  chapters,  deal  with  hitherto  unpublished  results. 


CHAPTER  I 

TRAJECTORIES  IN  AN  ARBITRARY  FIELD  OF  FORCE 
§§  1-8.     TRAJECTORIES  IN  THE  PLANE 

1.  We  consider  first  the  motion  of  a  particle  in  the  plane 
under  the  action  of  any  positional  field  of  force.  The  general 
equations  of  motion  are 

d  zc  d  TI 

m'dj?  =  ^X)  y^'    m  W  =  ^X'  ^' 

where  m  is  the  mass  and  <p,  \l/  are  the  rectangular  components 
of  the  force  acting  at  any  point  x,  y.  There  is  no  loss  of  gener- 
ality in  assuming  the  mass  of  the  particle  to  be  unity,  so  we  write* 

(1)  x  =  <p(x,  y),        y  =  $(x,  y). 

The  particle  may  be  started  from  any  position  (XQ,  yQ)  with 
any  velocity  (A0,  y0).  A  definite  trajectory  is  then  described. 
Since  the  same  curve  may  be  obtained  by  starting  from  any  one 
of  its  <x> x  points,  the  total  number  of  trajectories,  for  all  initial 
conditions,  is  oo3.  The  differential  equation  of  the  third  order 
representing  this  system  of  trajectories,  found  by  eliminating 
the  time  from  (1),  is  . 

(2)  y,  -  y'<p}y'"  =   MX  +  (ft,  -  9x)y'  -  9yy*\y"  -  3^/"2. 

This  is  not  an  arbitrary  differential  equation  of  the  third  order. 
Hence  the  system  of  trajectories  generated  by  an  arbitrary  field 
of  force  must  have  peculiar  geometric  properties,  which  translate 
the  peculiar  analytic  form  of  (2). 

*  The  following  notation  is  employed  throughout  these  lectures:  Dots  indicate 
total  differentiation  with  respect  to  the  time  t;  primes  indicate  total  differ- 
entiation with  respect  to  x;  subscripts  x  and  y  indicate  partial  differentiation ; 
finally,  the  subscript  s  indicates  total  differentiation  with  respect  to  the  arc 
length  s. 

7 


8  THE   PRINCETON    COLLOQUIUM. 

2.  Before  stating  these,  we  remark  that  a  more  intrinsic  basis 
for  the  discussion  is  obtained  by  decomposing  the  acting  force 
into  components  normal  and  tangential  to  the  path,  instead  of 
parallel  to  x  and  y  axis  as  in  (1).  Denoting  these  components 
by  N  and  T  respectively,  the  equations  of  motion  are 

(3)  v*/r  =  N,    vvs  =  T, 

where  v  denotes  the  speed,  s  the  arc  length,  and  r  the  radius  of 
curvature.  By  differentiating  the  first  of  these  equations  with 
respect  to  s,  and  comparing  with  the  second  equation,  we  can 
eliminate  v,  obtaining 

(4)  (rtf).  =  2T, 

a  relation  which  defines  the  trajectories  and  is  equivalent  to  (2). 
To  reduce  this  to  a  more  explicit  form,  we  introduce  an  auxiliary 
vector,  completely  determined  by  the  given  field  of  force,  namely 
the  space  derivative  of  the  force  (considered  of  course  as  a 
vector).  The  normal  and  tangential  components  of  the  force 
vector  are 


T  = 


the  corresponding  components  of  the  new  vector  are 


Vi  +  /  1  + 


While  the  new  vector  is  the  s  derivative  of  the  force  vector,  its 
components  are  obviously  not  the  same  as  the  s  derivatives  of 
the  old  components:  the  correct  relations  are  found  to  be 

(7)  N.  =  91  -          Ta  =  X  +     . 


ASPECTS   OF  DYNAMICS. 

These  formulas  are  sufficient  for  the  discussion  of  trajectories.* 
By  means  of  (7)  we  can  reduce  (4)  to  the  form 

(8)  Nr.  =  —  Wr  +  3  T. 

\     / 

This  is  the  fundamental  intrinsic  representation  of  the  system  of  oo3 
trajectories  connected  with  a  given  field  of  force. 

From  it  we  may  derive  very  simply  a  number  of  geometric 
properties.  But  in  dealing  with  the  converse  questions  which 
arise,  and  in  proving  the  completeness  of  the  set  obtained,  it  is 
more  convenient  to  use  the  equivalent  cartesian  representation, 
that  is,  equation  (2).f 

3.  The  Five  Characteristic  Properties  in  the  Plane. — The  system 
of  trajectories  generated  by  any  positional  field  of  force  in  the 
plane  has  the  following  set  of  properties,  and  conversely,  any 
system  of  oo3  curves  which  has  these  properties  will  be  a  system 
of  dynamical  trajectories. 

*  In  some  of  the  later  discussions  we  shall  need  also  the  space  derivatives 
of  9f  and  %,  which  may  be  written  in  the  form 

9i2  £2 

where 

9fi  =  : — - — ^-~rn> 


_ 

•J+2     ~~ 


n 

!+!/'* 


d+y'2)3/2 


<t>u 

£2    =    ^ 


2  • 

1+2/'2 

The  functions  0,  \j/  depend  only  on  the  position  of  the  particle;  the  auxiliary 
intrinsic  functions  N,  T,  3i,  £,  9Ji,  9^2,  ^E],  ^2,  denned  above,  depend  also 
upon  the  direction  of  motion;  finally,  N,,  T,,  9J,,  £s  depend  upon  the  curvature 
of  the  path.  Cf.  Bull.  Amer.  Math.  Soc.,  vol.  15  (1909),  p.  475. 

t  Cf.  Trans.  Amer.  Math.  Soc.,  vol.  7  (1906),  pp.  401-424.  The  result 
contained  in  property  IV  of  §  3  gives  this  simple,  but  apparently  overlooked, 
dynamical  theorem:  If  a  particle  starts  from  rest,  the  initial  curvature  of  the 
path  described  is  one  third  of  the  curvature  of  the  line  of  force  through  the 
initial  position. 


10  THE   PRINCETON  COLLOQUIUM. 

I.  If  for  each  of  the  oo1  trajectories  passing  through  a  given 
point  in  a  given  direction  we  construct  the  osculating  parabola, 
at  the  given  point,  the  locus  of  the  foci  of  these  parabolas  is  a 
circle  passing  through  that  point. 

II.  The  circle  that  corresponds,  according  to  property  I,  to  a 
lineal  element,  is  so  situated  that  the  element  bisects  the  angle 
between  the  tangent  to  the  circle  and  a  certain  direction  fixed 
for  the  given  point  (the  direction  of  the  force  acting  at  the  given 
point). 

III.  In  each  direction  at  a  given  point  there  is  one  trajectory 
which  has  four-point  contact  with  its  circle  of  curvature:  the 
locus  of  the  centers  of  the  QO  l  hyperosculating  circles  constructed 
at  the  given  point  is  a  conic  passing  through  that  point  in  the 
fixed  direction  described  in  property  II. 

IV.  With   any   point   0   there   is  associated  a  certain  conic 
passing  through  it  as  described  in  property  III.     The  normal 
to  the  conic  at  0  cuts  the  conic  again  at  a  distance  equal  to  three 
times  the  radius  of  curvature  of  the  line  of  force  passing  through 
0.     (The  lines  of  force  are  defined  geometrically  by  the  fact 
that  the  tangent  at  any  point  has  the  direction  associated  with 
that  point  in  accordance  with  property  II.) 

V.  When  the  point  0  is  moved,  the  associated  conic  referred 
to  above  changes  in  the  following  manner.     Take  any  two  fixed 
perpendicular  directions  for  the  x  direction  and  the  y  direction; 
through  0  draw  lines  in  these  directions  meeting  the  conic  again 
at  A  and  B  respectively.     Also  construct  the  normal  at  0  meeting 
the  conic  again  at  A7.     At  A  draw  a  line  in  the  y  direction  meeting 
this  normal  in  some  point  A',  and  at  B  draw  a  line  in  the  x 
direction  meeting  the  normal  in  some  point  B'.     The  variation 
property  referred  to  takes  the  form 


_ 


3cu2 


where  AA'  and  BB'  denote  distances  between  points,  and  where 
co  denotes  the  slope  of  the  lines  of  force  relative  to  the  chosen 


ASPECTS   OF   DYNAMICS.  11 

x  direction.     This  is  true  for  any  pair  of  orthogonal  directions, 


FIG.  1. 

and  therefore  really  expresses  an  intrinsic  property  of  the  system 
of  curves. 

4.  The  most  general  system  of   oo3  curves  in  the  plane  is 
represented  by  an  arbitrary  differential  equation  of  the  third  order 

TO  y'"=f(x,y,y',y"}. 

It  thus  involves  one  arbitrary  function  of  four  arguments. 

A  system  of  dynamical  trajectories,  on  the  other  hand,   is 
represented  by  an  equation  of  the  particular  form 


<FV) 


and  thus  involves  two  arbitrary  functions  of  two  arguments. 
These  are  the  only  systems  having  all  five  properties  I-V. 

It  is  interesting  to  notice  just  how  the  successive  imposition 
of  the  properties  gradually  narrows  down  the  general  form  (F0) 
to  the  particular  form  (Fv). 

5.   The  most  general  system  having  property  I  is  found  to  be 

y'"=G(x,y,y'}y"+H(x,y,y'}y"\ 


12  THE   PRINCETON   COLLOQUIUM. 

It  thus  involves  two  arbitrary  functions  of  three  arguments. 
This  type  of  course  includes  the  dynamical  type  as  a  very  special 
case.  It  arises  in  a  number  of  different  geometric  and  physical 
investigations.  It  has  therefore  its  own  interest.  The  char- 
acteristic property  may  be  stated  in  various  ways,  all  of  course 
equivalent  to  the  original  form:  (I)  The  osculating  parabolas 
of  the  trajectories  passing  through  a  given  point  in  a  given 
direction  have  the  foci  situated  on  a  circle  passing  through  the 
given  point.  Five  equivalent  forms  are  as  follows: 

I  (2).  The  directrices  of  the  osculating  parabolas  form  a  pencil 
It  follows  that  there  exists  a  point_(the  vertex  of  this  pencil) 
from  which  all  the  parabolas  subtend  an  angle  of  90°. 

I  (3).  If  for  each  of  the  trajectories  considered,  we  construct 
the  center  of  curvature  of  its  evolute,  the  locus  of  the  centers 
thus  obtained  is  a  parabola  passing  through  0,  and  having  its 
axis  parallel  to  the  given  initial  direction. 

I  (4).  For  each  of  the  trajectories,  construct  the  osculating 
equiangular  spiral.  The  locus  of  the  centers  of  the  poles  of 
these  spirals  is  a  circle  passing  through  0. 

I  (5).  Construct  for  each  of  the  trajectories  the  axis  of  devia- 
tion, that  is  the  line  bisecting  the  chords  of  the  curve  which  are 
parallel  and  infinitesimally  close  to  the  tangent.  The  tangent  of 
the  angle  between  the  varying  axis  of  deviation  and  the  fixed 
normal  is  a  linear  function  of  the  radius  of  curvature. 

I  (6).  The  derivative  of  the  radius  of  curvature  with  respect 
to  the  arc  length  is  a  linear  integral  function  of  the  radius  of 
curvature.  This  is  practically  a  restatement  of  (5),  since  for 
any  curve  the  derivative  of  the  radius  of  curvature  is  known  to  be 
equal  to  three  times  the  tangent  of  the  angle  of  deviation.  But 
in  this  form  it  has  the  advantage  of  being  valid,  not  only  in  the 
plane,  but  in  space  of  three  and  in  fact  any  number  of  dimensions. 

If  in  addition  to  property  I,  we  impose  property  II,  the  function 
H(x,  y,  y')  is  specialized  to 

77  =  y'  -  «(*,  y)' 


ASPECTS   OF   DYNAMICS.  13 

Thus  the  most  general  system  with  properties  I  and  II  is 

(Fn)  (y1  -  u}y'"  =  (y'  -  co) 


where  G  is  any  function  of  x,  y,  y',  and  w  is  any  function  of  x,  y. 
The  type  thus  involves  one  arbitrary  function  of  three  arguments 
and  one  arbitrary  function  of  two  arguments. 

6.  Systems  with  Properties  7,  II,  III.  —  Imposing  also  property 
III,  we  find  that  G(x,  y,  y'}  must  be  of  the  special  form 


Cr  =  -  -,  -  —  . 

y'  —  u 

Thus  the  most  general  system  of  oo3  curves  having  properties  I,  II, 
III  is  represented  by 

<fm)        (yf  -  «)y'"  =  (V2  +  \tf  +  ")</"  +  3i/"2, 

involving  four  arbitrary  functions  co,  X,  /z,  v  each  of  the  two 
arguments  x,  y. 

This  type  may  be  characterized  by  the  following  properties 
which  are  then  equivalent  to  I,  II,  III. 

I  (2).  For  a  given  lineal  element,  the  directrices  of  the   oo1 
osculating  parabolas  pass  through  a  common  point  D. 

II  (2).  When  the  lineal  element  turns  about  the  given  point  0, 
the  point  D  describes  a  straight  line  passing  through  0* 

III  (2).  The  correspondence  between  the  range  of  points  D 
and  the  pencil  of  elements  through  0  is  one-to-two  of  the  special 
form 

3 
~-j  =  X  sin2  6  +  n  sin  6  cos  6  +  v  cos2  6, 

where  d  denotes  the  distance  OD,  and  6  is  the  angle  between  the 
element  and  fixed  direction  of  OD. 


*  In  the  dynamical  case  this  line  OD  is  perpendicular  to  the  force  vector 
acting  at  0.  For  certain  special  fields  the  point  D  may  remain  fixed:  this 
happens  only  when  the  components  of  the  force  are  conjugate  harmonic 
functions,  that  is  when  the  field  is  of  the  type  termed  "  analytic  "  by  Lecornu. 


14  THE   PRINCETON   COLLOQUIUM. 

7.  If  now  we  add  the  properties  IV  and  V,  the  four  functions 
co,  X,  ju,  v  appearing  in  (-Fm)  must  obey  the  relations 

(/Yv)  Xco2  +  M^  H~  v  +  u>x  +  coco,,  =  0, 

(/V)  K  +  Xco  +  M)*  -  Xx  =  0. 

Thus  the  general  system  having  properties  I-IV  involves  three 
arbitrary  functions  of  x,  y;  while  that  having  all  five  properties 
involves  two  such  functions. 

By  integrating  these  relations,  we  may  express  the  four  functions 
in  terms  of  two  arbitrary  functions  <p,  \f/  as  follows  : 

*'         <f>V  — 

X=  —  , 


<P  <f>  <P  <P 

These  values,  substituted  in  the  type  (Fin),  actually  give  rise 
to  the  type 


and  thus  the  proof  is  completed  that  the  set  of  five  properties 
characterizes  the  dynamical  type. 

In  connection  with  the  statements  I  (2),  II  (2),  III  (2),  property 
IV  may  be  formulated  as  follows: 

IV  (2)  .  In  the  correspondence  described  in  III  (2)  ,  if  the  element 
approaches  the  direction  of  the  force  the  corresponding  distance 
OD  has  for  its  limiting  value  3/2  the  radius  of  curvature  of  the 
line  of  force  passing  through  0.  It  is  to  be  remembered  that 
the  direction  of  the  force,  and  hence  also  the  lines  of  force,  are 
defined  purely  geometrically  in  terms  of  the  given  triply  infinite 
system  of  curves  by  the  fact  that  at  any  point  0  in  the  plane 
the  "  direction  of  the  force  "  is  perpendicular  to  the  line  de- 
scribed as  the  locus  of  D  in  the  above  equivalent  II  (2)  of 
property  II. 

In  the  same  line  of  ideas  it  would  be  possible  to  find  an  equiva- 
lent for  property  V  (thus  completing  the  characterization),  but 
the  result  V  (2)  cannot  be  put  into  simple  form.  The  original 
form  V  may  be  criticized  as  inelegant  because  in  it  reference 
is  made  to  a  svstem  of  cartesian  axes.  Of  course  the  result 


ASPECTS   OF  DYNAMICS.  15 

expresses  an  intrinsic  property  since  it  is  true  for  all  systems  of 
axes.  It  would  certainly  be  desirable  to  restate  the  result  in 
intrinsic  language.  This  can  be  done,  for  instance,  by  introducing 
the  distances  cut  off  by  the  conic  described  in  IV,  not  only  on 
the  normal  ON,  but  also  on  the  two  lines  inclined  at  an  angle  of 
45°  to  that  normal.  However  it  does  not  seem  possible  to  obtain 
a  statement  which  is  both  simple  and  intrinsic  in  form. 

8.  Of  course  many  other  properties  of  trajectories  may  be 
obtained,  either  by  reasoning  synthetically  from  the  five  funda- 
mental properties,  or  by  reasoning  analytically  from  the  funda- 
mental differential  equation.  We  state  only  a  few  samples. 

If  we  shoot  particles  from  a  given  position  in  a  given  .direction 
with  variable  speed,  the  center  of  curvature  of  the  resulting 
trajectories  describes  a  straight  line  (the  normal)  and  the  focus 
of  the  osculating  parabola  simultaneously  describes  a  circle 
(by  property  II),  in  such  a  way  that  the  two  ranges  (one  linear, 
the  other  circular)  are  homographically  related;  furthermore  the 
given  point,  which  is  on  both  ranges,  corresponds  to  itself. 

If  we  shoot  from  the  same  position  in  a  direction  perpendicular 
to  that  previously  employed,  the  new  focal  circle  will  be  tangent 
(at  the  given  point)  to  the  former  focal  circle.  Conversely  if 
two  focal  circles,  for  the  same  point,  are  tangent,  the  initial 
directions  to  which  they  correspond  will  be  perpendicular  to 
each  other. 

We  shall  make  use  of  the  following  properties  which  describe 
the  disposition  of  the  oo1  focal  circles  constructed  at  a  given 
point.  The  two  results  which  follow  are  geometrically  equivalent, 
and  either  may  be  substituted  for  property  III  in  the  fundamental 
set. 

Ill  (3).  If  for  each  of  the  elements  at  a  given  point  we  construct 
the  corresponding  focal  circle,  the  locus  of  the  centers  of  the  oo1 
circles  thus  obtained  is  a  conic  with  one  focus  at  that  point. 

Ill  (4).  The  envelope  of  the  oo1  focal  circles  is  always  a  circle.* 

*  This  enveloping  circle  is  in  general  position :  it  does  not  usually  have 
its  center  at  the  given  point.  This  simple  position  arises  only  when  the  force 
is  of  the  Lecornu  type. 


16  THE  PRINCETON  COLLOQUIUM. 

§  9.    ACTUAL  AND  VIRTUAL  TRAJECTORIES 

9.  If  we  consider  the  motion  of  a  cannon  ball  in  a  given  vertical 
plane  under  the  action  of  gravity  assumed  constant,  the  triply 
infinite  system  of  trajectories  consists  of  parabolas  with  vertical 
axes.  We  do  not,  however,  obtain  all  vertical  parabolas,  rep- 
resented by  the  differential  equation  of  the  system  of  trajectories, 
which  is  here  y'"  =  0,  but  only  those  whose  concavity  is  directed 
downwards.  The  other  vertical  parabolas,  with  concavity 
directed  upwards,  satisfy  the  same  differential  equation,  and 
it  is  therefore  convenient  to  include  them  in  the  system  studied. 
We  thus  have  a  distinction  of  actual  and  virtual  trajectories. 
The  latter  are  the  actual  trajectories  corresponding  to  gravity 
reversed  in  direction. 

In  an  arbitrary  field  of  force  the  same  distinction  arises. 
The  complete  system  of  trajectories  is  composed  of  the  actual 
trajectories  corresponding  to  the  given  force,  and  the  virtual 
trajectories  which  are  the  actual  trajectories  corresponding  to 
the  reversed  field.  It  is  obvious  that  the  system  of  trajectories 
is  not  changed  if  the  force  acting  at  every  point  is  multiplied 
by  a  constant.  If  we  were  considering  only  actual  trajectories, 
it  would  be  necessary  to  restrict  this  constant  to  positive 
values,  but  as  we  include  both  actual  and  virtual,  the  constant 
factor  may  also  be  negative.  (Of  course  the  constant  must 
not  be  zero,  since  then  the  force  would  vanish  and  we  should 
obtain  the  trivial  system  of  straight  lines.) 

It  is  easy  to  show  that  the  virtual  trajectories  corresponding 
to  the  given  field  may  be  found  by  giving  the  initial  speed  of 
the  particle  a  pure  imaginary  value.  The  cannon  ball  could  be 
made  to  describe  a  parabola  with  its  concavity  directed  upwards 
if  only  some  kind  of  powder  could  be  invented  which  would 
cause  its  initial  speed  to  be  imaginary! 

In  discussing  the  general  geometric  properties  of  trajectories, 
we  had  in  mind  of  course  the  complete  system  as  defined  by  the 
differential  equation.  Consider  for  example  property  I:  for  any 


ASPECTS   OF  DYNAMICS.  17 

given  lineal  element  the  locus  of  the  foci  of  the  parabolas  oscu- 
lating the  corresponding  trajectories  is  a  circle  through  the  given 
point.  The  question  arises,  what  part  of  this  circle  corresponds 
to  the  actual  trajectories.  It  is  easily  found  to  be  the  arc  of  the 
circle  cut  off  by  the  initial  direction  line  (the  common  tangent 
of  the  trajectories  considered)  on  that  side  which  is  indicated 
by  the  force  vector.  Thus,  if  we  confined  our  discussion  to 
actual  trajectories,  the  focal  locus  would  be,  not  a  circle,  but  an 
arc  of  a  circle,  the  arc  running  from  the  given  point  0  to  a  certain 
terminal  point  A.  If  we  consider  all  elements  through  0  the 
locus  of  the  corresponding  terminal  point  A  is  found  to  be  a 
conic  passing  through  0  in  the  direction  of  the  force  vector.* 

For  a  given  element,  the  point  A,  which  separates  the  actual 
from  the  virtual,  may  be  defined  as  the  limiting  position  of  the 
focus  of  the  osculating  parabola  as  the  initial  speed  becomes 
infinitely  large.  The  osculating  parabola  in  this  limiting  case 
becomes  a  straight  line,  but  the  focus  has  a  definite  limiting 
position. 

An  analogous  distinction,  into  actual  and  virtual,  presents  itself 
also  in  the  theories  of  brachistochrones,  catenaries,  and  tau- 
tochrones.  The  differential  equations  of  the  systems  of  curves 
are  satisfied  by  both  the  actual  and  virtual  curves,  and  it  is  the 
complete  systems  that  we  refer  to  in  all  our  discussions  unless 
the  contrary  is  explicitly  mentioned. 

§§  10-15.    TRAJECTORIES  IN  SPACE 

10.  Consider  the  motion  of  a  particle,  which  we  may  take  to 
be  of  unit  mass,  in  an  arbitrary  positional  field  of  force.  The 
equations  of  motion  are 

(1)      x  =  <p(x,  y,  z),        y  =  \l/(x,  y,  z),        z  =  x(x,  y>  «)• 

The  particle  may  be  started  from  any  position,  in  any  direction, 
with  any  speed:  its  motion  is  then  determined  by  the  field  of 

*  This  conic  is  not  the  same  as  the  conic  arising  in  property  III. 
10 


18  THE   PRINCETON    COLLOQUIUM. 

force,  and  it  describes  a  definite  trajectory.  The  totality  of 
trajectories  constitutes  a  definite  system  of  oo5  curves.  (We 
exclude  the  case  where  the  force  vanishes  at  every  point,  the 
trajectories  then  being  merely  the  oo4  straight  lines.) 

What  are  the  properties  of  such  quintuply  infinite  systems  of 
curves?  Obviously  an  arbitrary  system  of  space  curves  cannot 
be  obtained  as  the  totality  of  trajectories  connected  with  any 
field  of  force.  In  fact  the  most  general  system  of  oo5  curves 
(assuming  that  oo1  curves  pass  through  any  point  of  space  in 
any  direction)  would  be  represented  by  a  pair  of  differential 
equations,  one  of  the  third  order  and  one  of  the  second  order, 
of  the  general  form 

(2)     y'"  =  f(x,  y,  z,  y',  z',  y"},      z"  =  g  (x,  y,  z,  y',  z',  y"), 

thus  involving  two  arbitrary  functions  of  six  arguments;  while 
the  dynamical  type  involves  merely  three  arbitrary  functions  of 
three  arguments.  The  differential  equations  representing  the 
dynamical  type,  obtained  by  eliminating  the  time  from  the 
equations  of  motion,  may  be  written  in  the  form 

1       <Pz  +  y '<Py  4~  Z'<( 

(t  -  y'v)z"  =  (x-  z'<p}y". 

The  question  is  to  express  the  peculiar  form  of  these  equations 
in  simple  geometric  language. 

The  interpretation  of  the  second  equation  is  obvious;  the  os- 
culating plane  of  the  path  passes  not  only  through  the  given 
initial  direction  1  :  y'  :  z',  but  also  through  the  fixed  direction 
<p  :\f/  :  x',  that  is,  the  osculating  plane  always  passes  through  the 
direction  of  the  force  acting  at  the  given  point.  The  other 
properties  are  not  obvious;*  they  take  into  account  the  form  of 
the  differential  equation  of  third  order. 

*  The  simplest  of  these,  property  II  below  and  certain  consequences,  were 
first  stated  in  the  author's  note  published  in  the  Bull.  Amer.  Math.  Soc., 


ASPECTS   OF   DYNAMICS.  19 

We  cannot  now,  as  in  the  case  of  the  plane  discussion,  employ 
osculating  parabolas,  since  our  curves  are  twisted.  Three 
consecutive  points  of  a  curve  determine  an  osculating  circle. 
What  do  four  consecutive  points  determine?  No  simple  type 
of  osculating  curve  is  available,  so  we  shall  make  use  of  the 
osculating  sphere.  The  results  are  therefore  quite  different  in 
form  from  those  obtained  in  the  two-dimensional  theory. 

11.  The  Four  Properties  in  Space. — In  order  that  a  system  of 
oo5  space  curves,  of  which  oo1  pass  through  each  point  in  each 
direction,  shall  be  identifiable  with  the  system  of  trajectories 
generated  by  a  positional  field  of  force,  it  is  necessary  and  suffi- 
cient that  it  shall  have  the  following  four  purely  geometric 
properties : 

I.  The  osculating  planes  of  the  oo3  curves  passing  through  a 
given  point  form  a  pencil;  that  is,  all  the  planes  pass  through  a 
fixed  direction. 

II.  The  osculating  spheres  of  the  oo1  curves  passing  through 
a  given  point  in  a  given  direction  form  a  pencil;  their  centers 
thus  lie  on  a  straight  line. 

III.  The  straight  lines  which  correspond,  in  accordance  with 
II,  to  all  the  oo 2  directions  at  a  given  point,  form  a  congruence 
(of  order  one  and  of  class  three)  consisting  of  the  secants  of  a 
twisted  cubic  curve;  which  cubic  furthermore  passes  through  the 
given  point  in  the  direction  fixed  by  property  I. 

IV.  The  associated  plane  systems  S',  determined  by  the  given 
space  system  in  the  manner  described  below,  have  the  five  geo- 
metric properties  characteristic  of  a  system  of  plane  dynamical 
trajectories.     Consider  the  given  system  of  oo5  space  curves  in 
connection  with  any  plane  p.     Through  any  point  of  p  there  pass 
oc2  curves  of  the  given  system  which  are  tangent  to  the  plane. 
Project  the  differential  elements  of  the  third  order  belonging  to 
these   space   curves   orthogonally   upon  p,   thus   obtaining    oo2 

vol.  12  (1905),  pp.  71-74.  Somewhat  simplified  proofs  were  then  given  by 
Cesaro,  in  a  paper  published  shortly  before  his  death,  in  the  Memorie  di 
Torino  (1905).  The  complete  set  of  properties  appeared  in  the  Trans.  Amer. 
Math.  Soc.,  vol.  8  (1907),  pp.  121-140. 


20  THE   PRINCETON   COLLOQUIUM. 

plane  differential  elements  of  the  third  order  at  the  selected  point. 
Applying  this  process  to  all  points  of  p,  we  have  a  defined  set 
of  oo 4  differential  elements  of  the  third  order.  These  elements 
define  a  certain  differential  equation  of  the  third  order,  and  thus 
determine  a  system  of  oo3  integral  curves.  This  we  term  the 
associated  system  in  the  plane  p.  The  space  system  has  the 
property  that  every  one  of  these  plane  systems  associated  with 
it  is  a  system  of  dynamical  trajectories,  and  therefore  has  the 
five  properties  stated  in  §  3,  which  we  here  denote  by  IP-VP 
in  order  to  avoid  confusion  with  the  four  spatial  properties. 

These  four  properties  are  ordinally  independent:  no  one  can 
be  derived  from  those  which  precede  it.  The  question  of  absolute 
independence  is  left  open:  it  is  quite  probable  that  IV  is  suf- 
ficiently strong  to  furnish  a  complete  characterization  by  itself. 

12.  The  most  general  system  having  property  I  involves  one 
arbitrary  function  of  six  arguments  besides  two  functions  of 
three  arguments.     These  systems  have  the  following  properties, 
which  are  of  course  consequences  of  property  I. 

The  oo1  curves  passing  through  a  given  point  in  a  given  di- 
rection have  not  only  the  same  osculating  plane,  but  also  the 
same  torsion. 

If  the  torsion  is  given  the  corresponding  initial  directions  form 
a  quadric  cone.  In  particular  such  a  cone  defines  the  directions 
of  those  curves,  through  the  given  point,  which  admit  hyper- 
osculating  planes. 

If  for  each  of  these  curves  we  construct,  at  the  common  point, 
the  related  helix*  (that  is  the  helix  which  agrees  with  the  curve 
in  osculating  plane,  curvature,  and  torsion),  the  axes  of  the 
helices  so  obtained  generate  a  cylindroid. 

13.  The  most  general  system  with  properties  I  and  II  involves 
two  arbitrary  functions  of  five  arguments,  besides  two  functions 
of  three  arguments.     Two  further  statements,  each  equivalent 
to  II,  are  as  follows: 


*An  osculating  helix,  that  is,  one  having  four-point  contact  with  the  curve, 
does  not  in  general  exist. 


ASPECTS   OF  DYNAMICS.  21 

If  for  each  of  the  oo1  curves  defined  by  a  given  lineal  element 
we  construct  the  osculating  circle  and  the  osculating  sphere, 
the  distance  between  the  center  of  the  circle  and  the  center  of 
the  sphere  varies  as  a  linear  integral  function  of  the  radius  of 
curvature. 

For  the  same  set  of  oo1  curves,  the  derivative  of  the  radius  of 
curvature  with  respect  to  the  arc  length  can  be  expressed  as  a 
linear  integral  function  of  the  radius  of  curvature. 

This  last  form  has  the  advantage  of  being  valid  in  space  of 
two  or  any  number  of  dimensions.  On  this  basis,  however,  it 
would  be  difficult  to  formulate  equivalents  for  the  higher  prop- 
erties, so  as  to  obtain  a  complete  characterization. 

14.  Property  III  is  perhaps  the  most  interesting  result  obtained. 
The  most  general  system  having  this  property  in  addition  to  I 
and  II  is  represented  by  a  pair  of  differential  equations  involving 
ten  arbitrary  functions  of  three  arguments. 

One  may  ask  what  is  the  significance  of  the  cubic  curve 
(we  denote  it  by  F),  which  arises  in  connection  with  III.  To 
each  point  0  of  space  there  is  related  a  certain  cubic  F.  If  we 
shoot  from  0  in  every  direction  with  every  speed,  we  obtain  oo3 
trajectories.  Each  of  these  has  an  osculating  sphere  with  a 
definite  center  C.  To  each  of  the  trajectories  there  corresponds 
one  center  C.  Usually  the  center  C  determines  the  trajectory. 
However  if  C  lies  on  the  curve  F,  there  are  oo1,  instead  of  one, 
corresponding  trajectory:  in  this  case  in  fact  the  initial  direction 
may  be  any  direction  perpendicular  to  the  line  joining  0  and  C.* 
Thus  the  curve  F  may  be  defined  as  the  locus  of  those  points 
which  may  serve  as  centers  for  more  than  one  trajectory  through 
the  given  point  0. 

A  simple  consequence  of  III  is  that  the  locus  of  the  centers  of 
the  osculating  spheres  of  the  oo2  trajectories  touching  a  given 
plane  at  a  given  point  is  a  quadric  surface. 


*  Two  trajectories  through  0  have  the  same  osculating  sphere  only  if  the 
initial  speed  is  the  same,  and  if  the  line  through  0  perpendicular  to  the  initial 
elements  meets  the  cubic  r. 


22  THE   PRINCETON   COLLOQUIUM. 

If  the  plane  varies,  the  given  point  being  held  fixed,  the  oo2 
quadrics  obtained  form  a  linear  system.* 

The  properties  so  far  considered  relate  to  the  curves  through  a 
given  point  0.  If  we  have  oo3  curves  passing  through  a  point  0, 
oo1  in  each  direction,  and  if,  at  that  point,  properties  I,  II,  III 
are  fulfilled,  it  will  not  usually  be  possible  to  generate  the  curves 
as  trajectories  in  any  field  of  force.  All  that  follows  is  that  the 
relations  between  y',  z',  y",  z",  y'",  z"r  are  of  precisely  the  same 
form  as  those  holding  for  trajectories;  and  therefore  it  is  possible 
to  find  (in  infinitely  many  ways)  a  field  of  force  such  that  each 
of  the  oo3  trajectories  passing  through  the  given  point  shall 
have  contact  of  the  third  order  with  some  one  of  the  given  curves. 

15.  In  order  to  cause  our  system  to  be  of  the  dynamical  type, 
it  is  necessary  to  restrict  the  ten  arbitrary  functions  involved 
in  the  type  characterized  by  I,  II,  III  so  that  only  three  arbi- 
traries  remain,  namely,  the  components  <p,  \f/,  x  defining  the  field 
of  force.     This  is  the  role  of  property  IV,  which  states  that  in 
any  plane  p  the  associated  system  5  is  of  the  plane  dynamical 
type.     An  equivalent  statement  is  as  follows: 

IV  (2).  If  the  oo2  space  curves  touching  any  plane  p  at  any 
point  0  are  projected  orthogonally  upon  p,  the  plane  curves  thus 
obtained  possess  the  properties  Ip,  IIP,  IIIP;  when  the  point  0 
varies  in  p,  the  direction  associated  with  it  by  IIP,  and  the  conic 
associated  with  it  by  IIIP,  vary  in  accordance  with  the  restrictions 
expressed  in  IVP  and  Vp. 

It  may  be  remarked  that  the  first  half  of  this  statement  holds 
for  all  space  systems  having  properties  I,  II,  III;  in  fact  all 
such  systems  have  also  property  IVP.  The  real  restriction  is 
in  Vp.  It  is  also  sufficient  to  consider,  instead  of  all  planes  p, 
merely  those  of  a  triply  orthogonal  set. 

§§  16-25.    THE  INVERSE  PROBLEM  OF  DYNAMICS:  A  METHOD 
OF  GEOMETRIC  EXPLORATION 

16.  The  usual  direct  and  inverse  problems  arising  in  dynamics 
are:  first,  given  the  force  acting  on  a  particle,  to  find  its  motion; 

*  On  the  other  hand  if  we  vary  the  given  point,  keeping  the  plane  fixed, 
no  simple  result  is  obtained:  the  oo2  quadrics  constitute  an  arbitrary  family. 


ASPECTS   OF   DYNAMICS.  23 

and  second,  given  the  motion  of  a  particle,  to  find  the  force 
acting  on  it.  The  first  problem  is  solved  by  integrating  the 
differential  equations  of  motion.  The  second  is  solved  by  dif- 
ferentiating the  coordinates  of  the  point  with  respect  to  the  time. 

Suppose,  however,  that  we  are  given  only  the  path  described 
by  the  particle  but  have  no  information  about  the  motion  along 
the  curve.  If  merely  a  single  curve  is  given,  the  problem  of 
finding  the  acting  force  would  of  course  be  indeterminate.  But 
if  all  the  trajectories,  described  by  starting  particles  in  a  field 
of  force  from  all  initial  positions  in  all  directions  with  all  speeds, 
are  given,  then  the  field  of  force  is  essentially  determined  (that 
is,  up  to  a  constant  factor).  Hence  if  ice  were  given  a  photograph 
of  the  entire  system  of  curves  generated  by  some  (positional)  field 
of  force,  without  any  record  of  motion  or  time,  it  ought  to  be  possible 
to  find  the  law  of  the  field  of  force.  This  is  easily  seen  to  be  true 
analytically;  but  we  wish  also  a  purely  geometric  solution 
which  will  enable  us  to  pass  from  the  given  curves  to  the  vector 
representing  the  force  at  each  point  of  the  plane  (taking  first 
the  two-dimensional  case).  The  result  gives  what  may  be 
described  as  a  method  for  the  geometric  exploration  of  a  field  of 
force. 

17.  First  consider  two  trajectories  passing  through  the  same 
point  0  in  the  same  direction.  Construct  the  two  osculating 
parabolas.  The  circle  passing  through  the  point  0  and  the  foci 
of  these  parabolas  will,  according  to  property  I,  be  the  focal 
circle  corresponding  to  the  given  point  and  the  given  direction. 
Then,  according  to  property  II,  the  direction  of  the  force  acting 
at  0  will  be  symmetric  to  the  tangent  to  this  circle  at  0  with 
respect  to  the  common  tangent  of  the  two  curves.  An  equivalent 
of  this  construction  is  to  join  0  to  the  intersection  of  the  direc- 
trices of  the  osculating  parabolas:  this  line  is  perpendicular  to 
the  direction  of  the  force  acting  at  0. 

If  we  are  given  two  trajectories  passing  through  0  in  different 
directions,  then  the  direction  of  the  force  at  0  is  not  determined. 
The  same  is  true  if  we  are  given  three  curves  with  distinct 
tangents. 


24  THE  PRINCETON   COLLOQUIUM. 

18.  //,    however,   we  are  given  four  trajectories   with  distinct 
tangents,  the  force  direction  is  (in  general)  uniquely  determined. 

Consider  an  arbitrary  direction  at  0,  and  let  us  see  if  it  can 
be  the  direction  of  the  force  acting  at  that  point.  Take  the 
image  of  this  direction  in  the  tangent  to  the  first  of  the  given 
curves;  then  pass  a  circle  through  0  in  the  direction  so  obtained 
and  through  the  focus  of  the  corresponding  osculating  parabola. 
Doing  this  for  each  of  the  four  curves,  we  obtain  four  focal  circles. 
//  there  exists  a  circle  touching  these  four,  the  direction  tested  is  cor- 
rect. This  follows  from  property  III  (4)  of  §  8.  We  have  then  a 
purely  geometric  problem:  to  find  a  direction  at  0  such  that  the 
four  circles  constructed  by  means  of  it  shall  admit  a  common 
tangent  circle.  We  may  simplify  this  problem  by  inverting  the 
configuration  considered  with  respect  to  0.  We  then  have, 
instead  of  the  four  focal  circles,  four  straight  lines  which  are  to 
be  concyclic.  As  we  change  the  direction  tested,  these  rotate 
simultaneously  through  equal  angles  about  four  fixed  points, 
namely,  those  obtained  by  inverting  the  four  foci. 

Take  an  arbitrary  oriented  direction  for  trial ;  construct  for 
each  of  the  four  inverse  foci,  a  direction  parallel  to  the  image  of 
the  tangent  to  the  focal  circle  with  respect  to  the  tangent  to 
the  corresponding  trajectory.  W7e  thus  obtain  four  oriented 
lineal  elements,  one  at  each  of  the  inverse  foci.  The  problem  is 
then  to  rotate  these  through  the  same  angle  a,  so  that  the  new 
elements  shall  have  concyclic  lines.*  In  this  position  the  image 
of  the  direction  of  any  one  of  the  four  elements  in  the  corre- 
sponding tangent  at  0  will  give  the  required  direction  of  the 
force.  The  only  ambiguity,  in  general,  will  be  in  the  sense  (arrow- 
head) of  the  force:  this,  however,  may  be  determined  separately 
for  actual f  trajectories  by  considerations  of  concavity  and  con- 
vexity. 

19.  The  direct  analytical  treatment  is  as  follows.     The  dif- 
ferential equation  of  the  oo3  trajectories  of  any  positional  field 

*  A  simple  ruler  and  compass  solution  of  this  problem  was  suggested  to 
the  author  by  Professor  Wedderburn. 
tSee§9. 


ASPECTS   OF  DYNAMICS. 

of  force  is  of  the  form 


25 


where  X,  /z,  v,  co  are  functions  of  x,  y  (and  have  therefore  fixed 
numerical  values  so  far  as  we  deal  with  the  oo2  curves  passing 
through  a  given  point  0),  the  latter  quantity  co  representing  the 
slope  of  the  acting  force.  Each  of  the  four  given  curves  Ci,  C2, 
(?3,  (74  through  the  point  0  determines  certain  values  or  the 
derivatives  y',  y",  y'";  that  is  we  are  given  the  differential  ele- 
ments of  third  order 

yi',    yi",    yi'"  (i  =  1,  2,  3,  4). 

Substituting  these  values  we  have  four  linear  equations 
(yi'  -  <*W  =  W  +  Mi'  +  vW  +  3y/'2     (i  =  1,  2,  3,  4), 

from  which  we  can  find  the  values  of  X,  ju,  v,  co  at  the  given  point. 
The  required  direction  of  the  acting  force  is  determined  by  the  slope 


co  = 


where  numerator  and  denominator  are  determinants  of  the  fourth 
order. 

20.  By  any  of  these  methods  we  may  determine  the  direction* 
of  the  vector  representing  the  force  acting  at  any  point  0  of  the 
plane.  How  shall  we  determine  the  magnitude  of  the  vector? 
The  determination  cannot  be  absolute,  since,  as  already  remarked, 
two  fields  that  differ  by  a  constant  factor  have  identical  trajec- 
tories. The  magnitude  of  the  vector  at  any  one  point  may 
be  taken  at  random,  and  then  the  field  is  completely  determined. 

This  depends  upon  the  simple  fact  that  if  we  know  the  path 

*  Of  course  if  all  the  trajectories  were  given,  the  direction  of  the  force 
would  be  determined  immediately  by  the  fact  that  the  curves  in  that  direc- 
tion have  zero  curvature. 


26  THE    PRINCETON   COLLOQUIUM. 

of  a  particle  and  also  the  direction  of  the  force  acting  at  each  of 
its  points,  then  assuming  the  magnitude  arbitrarily  at  one  point, 
it  is  completely  determined  at  all  points.  This  is  an  integration 
problem.  We  know  the  force  vector  at  the  initial  point  0,  and 
may  decompose  it  into  components  N  and  T,  normal  and  tangent 
to  the  given  curve.  Assuming  the  mass  to  be  unity,  the  initial 
speed  is  given  by 

v<?  =  rN, 

where  r  is  the  known  radius  of  curvature.     Then  from 

OT.  =  T, 

we  may  find  va,  the  rate  of  variation  of  the  speed  for  unit  of  arc. 
The  speed  at  any  point  P  of  the  curve  is  thus  found  in  the  form 


.  = 


where  all  the  quantities  in  the  right-hand  member  are  geo- 
metrically given.  (The  integrals  throughout  are  calculated  from 
point  0  to  point  P.)  If  we  denote  by  6  the  inclination  of  the 
force  to  the  curve,  so  that  tan  6  =  N/T,  the  speed  is 

/-cot  0 

I  -  —  ds 

v  =  v0e 

Since  the  speed,  that  is  the  motion,  is  now  known,  the  magni- 
tude of  the  force  is  also  known.  The  components  at  any  point  P 
are 

f-t  cot  t  —  r, 

N  =  NveJ  '  ,     T  =  N  cot  6. 


21.  We  see  that  the  construction  of  the  field  may  be  carried 
out  without  knowing  all  the  oo3  trajectories.  So  far  as  the 
direction  of  the  force  is  concerned,  it  is  sufficient  to  know  at 
each  point  of  the  plane  either  two  trajectories  with  a  common 
tangent,  or  four  trajectories  with  distinct  tangents.  So  far  as 


ASPECTS   OF   DYNAMICS.  27 

the  magnitude  is  concerned  it  is  then  sufficient  to  know  oo1  tra- 
jectories through  one  point  0,  one  for  each  direction,  since  we  can 
then  integrate  from  this  point  to  any  point  of  the  plane*  along 
some  one  of  the  curves. 

A  field  of  force  is  in  general  determined,  and  may  be  constructed, 
if  ice  know  4Q01  out  of  the  totality  of  oo3  trajectories,  each  of  the 
four  systems  of  oo1  curves  covering  the  plane  (or  the  region 
considered)  simply,  that  is,  one  passing  through  each  point  of 
the  plane. 

The  complete  system  of  oo3  trajectories  is  thus  determined  in 
general  by  four  systems  of  oo1  trajectories.  Further  reduction 
is  possible.  In  general  Sao1  curves  determine  the  totality,  but 
no  simple  constructions  are  then  available.  If  two  simply 
infinite  systems  of  curves  (that  is,  a  net  of  curves)  are  assigned 
arbitrarily,  a  corresponding  complete  system  can  be  found  in  a 
large  infinitude  of  ways:  the  corresponding  field  of  force  is  not 
determined  up  to  a  constant,  but  involves  arbitrary  functions. 

The  first  and  most  interesting  example  of  the  geometric  ex- 
ploration of  a  field  of  force  arose  in  Bertrand's  discussion  of 
Kepler's  laws.  The  first  of  these  laws  (every  planet  describes 
an  ellipse  having  the  sun  for  a  focus)  is  geometric,  while  the  second 
and  third  are  kinematic  (involving  the  areal  velocity  and  the 
period).  The  first  law  determines  all  the  trajectories,  and  there- 
fore determines  the  field  of  force.f  Hence  the  newtonian  law 
of  gravitation  can  be  deduced  from  the  first  law  alone,  instead  of, 
as  usual,  from  all  three.  Bertrand  thus  concludes  that  the 
other  two  laws  are  consequences  of  the  first.  If  Kepler  had  been 
a  mathematician  of  the  twentieth  century,  he  would  have  stopped 
his  laborious  observational  inductions  after  noting  his  first  law, 
and  deduced  the  other  two  analytically. 

The  first  law,  in  Bertrand's  discussion,  is  of  course  to  be  taken 
ideally:  not  only  the  actual  planets  describe  conies  with  a  focus 
at  the  sun,  but  every  particle  starting  from  any  position  with 

*  That  is,  in  some  region  of  the  plane — in  some  neighborhood  of  O. 
t  It  is  assumed,  of  course,  that  the  force  depends  only  upon  the  position 
of  the  planet. 


28  THE  PRINCETON  COLLOQUIUM. 

any  velocity  describes  such  a  conic.  From  what  has  been 
stated  above  it  is  sufficient  to  limit  the  observations  to  four 
simply  infinite  systems  of  conies  in  "  general  "  position. 

On  account  of  the  last  phrase,  it  is  easily  possible  to  commit 
errors  in  the  application  of  the  result.  It  would  be  possible  to 
give  4oo:  or  even  oo2  conies  in  certain  special  ways,  so  that  the 
field  is  not  determined.  (See  §  23.) 

22.  This  raises  the  general  question:  How  many  trajectories 
may  be  common  to  two  distinct  fields  of  force? 

The  first  field,  defined  by  its  components  <p,  $,  has  a  system  of 
oo3  trajectories  with  a  differential  equation 

(yf  -  co)/"  =  (X/2  +  w'  +  W  +  3/'2; 

the  second  field,  with  components  <pi,  \f/i,  has  a  system  of  tra- 
jectories given  by  an  analogous  equation 

(y'  -  co,)/"  =  (Xi/  +  MI/  +  W  +  3/'2. 


If  there  are  any  solutions  in  common,*  they  must  satisfy  the 
equation  of  second  order 

3(co  -  Wl)y"  =  (y1  -  co)(X^'2  +  Mi/  +  "i) 


Two  systems  of  trajectories  cannot  ham  more  than  oo2  curves 
(one  through  each  point  in  each  direction)  in  common  without 
coinciding.  If  they  have  oo2  curves  in  common  the  differential 
equation  of  the  second  order  defining  these  curves  must  be  of 
the  cubic  formf 

y"  =  Ay'3  +  By'2  +  Cy'  +  D, 

where  the  coefficients  are  functions  of  x,  y. 

Usually  the  solutions  of  the  equation  of  the  second  order  will 

*  In  addition  to  straight  lines,  y"  =  0,  which  are  common  to  all  systems. 

t  This  form  is  characterized  by  the  fact  that  the  locus  of  the  centers  of 
curvature  of  the  curves  passing  through  a  given  point  is  a  special  type  of 
cubic  curve.  Cf.  Amer.  Jour.  Math.,  1908,  p.  207. 


ASPECTS   OF  DYNAMICS.  29 

not  satisfy  either  equation  of  the  third  order  and  the  two  systems 
will  have  no  curves  in  common.  An  example  showing  that  the 
two  systems  may  actually  have  oo2  curves  in  common  is  given 
by  the  fields 

<p  =  X,      \f/  =  ty',  <pi  =  X~Z,      iff  i  =    1, 

where  the  equation  of  second  order, 

*y"  =  y', 

defines  oo2  curves  y  =  as?  +  b,  which  are  trajectories  in  both 
fields. 

23.  A  fortiori  4Q01  curves,  or  any  number  of  simple  systems, 
may  belong  to  two  distinct  fields.  If  the  four  simple  systems 
are  given  in  the  form 

y'  =  MX,  y}  (i  =  1,  2,  3,  4), 

the  field,  if  it  exists,  will  be  uniquely  defined  provided  not  all  the 
determinants  of  fourth  order  in  the  matrix 

1  1//,    W,    /#/,    /.-",     3//*-/<f.-"|| 

vanish  identically.  Here  the  primes  denote  complete  differen- 
tiation with  respect  to  x,  so  that 


"   =  fXX  +  2ffxy  +  f%y  +  fXfy  +  //„*. 


This  is  the  exact  formulation  of  the  result  stated  previously  "  in 
general." 

24.  Consider  the  simplest  of  all  fields,  gravity  assumed  constant. 
If  a  cannon  ball  is  projected  in  any  way  into  the  field  it  describes 
a  vertical  parabola.  Conversely  if  every  path  in  an  unknown 
field  is  a  vertical  parabola,  it  follows  that  the  acting  force  is 
vertical  and  constant  in  intensity.  How  many  cannon  ball 
experiments  would  haw  to  be  made  in  order  to  arrive  at  this  con- 
clusion? 

We  confine  the  discussion  for  simplicity  to  a  fixed  vertical 


30  THE   PRINCETON   COLLOQUIUM. 

plane,  taken  as  the  .ry-plane,  so  that  the  equations  of  motion  are 

x  =  0,         y  =  1 
and  the  trajectories  are  the  ce3  parabolas 

y  =  ax-  +  bx  +  c. 

Suppose  first  the  cannon  is  kept  in  one  place,  say  the  origin, 
and  the  ball  is  fired  in  all  directions  with  all  initial  speeds,  giving 
in  all  oc  -  parabolas 

y  —  ax2  -f-  bx. 

This  would  not  be  sufficient  to  prove  that  the  field  is  uniform. 
Another  possible  field,  for  example,  is 

x  =  x~5,        y  =  yx~6. 

In  fact  there  are  oo2  distinct  fields  each  consistent  with  the  given 
set  of  oo2  parabolas. 

The  same  is  true  if  we  confine  our  geometric  experiments  to 
the  oo 2  parabolas  y  =  ax-  +  c  found  by  shooting  horizontally 
from  every  point  in  the  axis  of  ordinates  with  variable  initial 
speed.  The  differential  equation  of  this  family  is  xy"  =  y', 
precisely  the  one  given  at  the  end  of  §  22,  and  so  the  two  forces 
there  given  are  consistent  with  the  experiments,  just  as  much 
as  ordinary  gravity. 

If  however  the  shots  are  fired  from  all  points  in  the  axis  of 
abscissas,  with  all  initial  speeds,  at  the  fixed  inclination  of  45°, 
producing  as  trajectories  the  oo2  vertical  parabolas  whose  foci 
are  on  the  axis  of  abscissas,  the  field  must  be  uniform  gravity. 
The  only  possible  field  is  in  fact  x  =  0,  y  =  constant. 

The  same  is  true  if  we  fix  the  amount  of  powder,  that  is  the 
initial  speed,  and  shoot  from  every  point  on  the  ground  (the 
axis  of  x),  at  every  angle.  This  gives  oo2  parabolas  with  a 
common  directrix. 

As  an  example  of  a  set  of  4Q01  observations  that  would  be 
sufficient,  we  mention  only  the  case  of  shooting  from  four* 

*  It  may  be  that  three  stations  are  sufficient,  but  this  requires  a  separate 
discussion.  Two  stations  would  certainly  not  suffice. 


ASPECTS   OF  DYNAMICS.  31 

stations  on  the  ground,  pointing  the  cannon  at  the  angle  45°, 
and  using  all  initial  speeds. 

25.  Consider  very  briefly  the  general  inverse  problem  in  space 
of  three  dimensions.     The  determination  of  the  magnitude  of 
the  force  involves  the  same  considerations  as  in  the  plane  case. 

If  we  are  given  two  trajectories  through  0  in  the  same  direction, 
the  osculating  planes  must  coincide.  The  force  acts  in  this 
common  plane;  its  direction  is  determined  by  projecting  the 
given  space  curves  orthogonally  on  this  plane,  and  then  using 
the  plane  construction  described  above. 

If  we  are  given  two  trajectories  with  distinct  osculating  planes, 
the  initial  directions  will  be  necessarily  distinct;  the  force-direc- 
tion is  then  determined  by  the  intersection  of  the  osculating 
planes. 

If  wre  are  given  two  trajectories  through  0  in  different  direc- 
tions, but  with  the  same  osculating  plane,  the  direction  of  the 
force  is  not  determined.  We  need  in  fact  four  such  curves  with 
the  same  osculating  plane  and  different  directions  before  the 
force-direction  is  determined:  the  requisite  construction  is  again 
obtained  by  orthogonal  projections  of  the  curves  of  the  common 
osculating  plane,  thus  reducing  the  problem  to  that  considered 
in  the  two-dimensional  theory  (cf.  §  18). 

§§    26-27.    TESTS  FOR  A  CONSERVATIVE  FIELD 

26.  Since  the  system  of  trajectories  determines  the  field  of 
force,  it  ought  to  be  possible  to  find  out  from  the  trajectories, 
whether  the  field  belongs  to  any  special  type,  for  example,  whether 
the  field  is  central  or  conservative. 

The  lines  of  force  are  determined  geometrically  by  property  I 
in  the  plane  and  property  II  in  space.  The  field  will  be  central  if 
the  lines  of  force  are  straight  lines  passing  through  a  common 
point. 

We  now  give  a  number  of  tests  any  one  of  which  will  distinguish 
a  conservative  from  a  non-conservative  force.  It  is  not  possible 
to  decide  this  from  the  lines  of  force  alone. 


32  THE   PRINCETON  COLLOQUIUM. 

1°.  First  consider  the  plane  theory.  Here  there  is  for  each 
point  a  certain  conic  determined  by  the  trajectories  in  accordance 
with  property  III  of  §  3  as  the  locus  of  the  centers  of  the  hyper- 
osculating  circles.  For  a  conservative  field  (and  for  no  other)  this 
conic  is  always  a  rectangular  hyperbola. 

2°.  In  connection  with  property  III  (3)  of  §  8  we  have  this  test: 
The  conic  which  there  appears  as  the  locus  of  the  centers  of 
the  focal  circles  is  in  the  conservative  case  merely  a  straight  line. 
That  is,  the  focal  circles  constructed  at  any  point  all  have  a 
second  point  in  common. 

3°.  The  focal  circles  corresponding  to  two  perpendicular 
directions  are,  in  any  field,  tangent  to  each  other.  In  the  con- 
servative case  the  two  circles  coincide. 

4°.  In  any  field  two  trajectories  through  a  given  point  0 
exist  whose  osculating  parabolas  have  the  same  given  focus. 
If  for  one  given  focus  the  trajectories  are  orthogonal  at  0,  this 
will  be  true  for  any  given  focus.  When  this  is  the  case  for  every 
point  0,  the  force  will  be  conservative. 

27.  In  the  three-dimensional  theory,  the  lines  of  force  in  the 
conservative  case  necessarily  form  a  normal  congruence;  but 
this  is  not  a  sufficient  test.  All  the  tests  given  below  are  both 
necessary  and  sufficient. 

1°.  First  consider  property  III  of  §  1 1.  In  any  field  there  corre- 
sponds to  each  point  0  a  certain  twisted  cubic  curve  F.  The  con- 
servative fields  are  distinguished  by  the  fact  that  the  cubic  F  is, 
for  every  point  0,  of  the  rectangular  type.* 

2°.  An  interesting  kinematic  test,  connected  with  the  theorem 
of  Thomson-Tait,  is  the  following.  If  from  any  point  0  we  shoot 
with  a  given  speed  r0  in  every  direction,  <x>2  trajectories  will  be 
obtained.  If  these  form  a  normal  congruence  (that  is  admit  a  set 
of  orthogonal  surfaces),  the  same  will  necessarily  be  true  for  any 
other  speed  r0.  The  trajectories  starting  out  from  any  -point  with 


*  That  is,  the  cubic  intersects  the  plane  at  infinity  in  three  mutually 
orthogonal  directions.  All  the  quadrics  passing  through  the  curve  are  then 
of  the  equilateral  type. 


ASPECTS  OF  DYNAMICS.  33 

a  given  speed  form  a  normal  congruence  when,  and  only  when,  the 
field  is  conservative. 

The  necessity  of  this  condition  is  included  in  the  Thomson- 
Tait  theorem  discussed  in  the  next  chapter.  Its  sufficiency,  of 
course,  requires  a  separate  discussion  which  is  connected  with 
the  theory  of  velocity  systems. 

3°.  In  order  to  make  the  preceding  test  purely  geometric,  it  is 
necessary  to  have  a  geometric  method  of  assembling  those  tra- 
jectories which,  starting  from  the  same  point,  correspond  to  the 
same  initial  speed.  Such  a  method  is  readily  found  from  the 
fact  that  the  square  of  the  speed  varies  directly  as  the  radius  of 
curvature  and  directly  as  the  normal  component  of  the  force. 
The  oo2  trajectories  corresponding  to  a  given  speed  have  circles 
of  curvature  intersecting  each  other  at  the  same  point  on  the 
line  of  the  force  vector;  that  is,  the  centers  of  curvature  lie  in  a 
plane  perpendicular  to  the  direction  of  the  force  acting  at  the 
given  point.  In  the  conservative  case,  the  oo2  trajectories  so 
selected  form  a  normal  congruence. 

4°.  Among  the  oo2  trajectories  considered  there  are,  for  any 
field,  three  wrhich  admit  hyperosculating  circles  of  curvature. 
The  three  initial  directions  thus  determined  will  be  mutually 
orthogonal  when  and  only  wrhen  the  field  is  conservative. 

Only  test  1°  is  directly  connected  with  the  set  of  properties 
I-IV  of  page  19.  The  other  three  are  suggested  by  the  discussion 
of  velocity  systems  (cf.  §  32). 


11 


CHAPTER  II 

NATURAL    FAMILIES:     THE    GEOMETRY    OF    CONSERVATIVE 
FIELDS  OF   FORCE 

§  28.    ORIGIN  AND  APPLICATION   OF  THE  NATURAL  TYPE 

28.  We  now  consider  the  properties  of  the  trajectories  gener- 
ated by  conservative  fields  of  force.  The  total  system  of  tra- 
jectories will  have  the  general  properties  previously  considered 
for  an  arbitrary  field  of  force,  together  with  the  additional  proper- 
ties stated  in  §§  26,  27,  peculiar  to  the  conservative  case. 

An  entirely  new  feature  presents  itself,  due  to  the  fact  that 
the  differential  equations  of  motion  admit  an  integral  of  the 
first  order,  namely,  the  energy  equation.  During  any  motion 
of  the  particle  in  the  given  field,  the  sum  of  the  kinetic  and 
potential  energies  is  constant;  thus  each  motion  corresponds 
to  a  definite  value  of  the  constant  h,  representing  the  total  energy. 
The  motions  may  therefore  be  grouped  according  to  the  values  of 
h.  Those  corresponding  to  a  given  value  form  what  may  be 
termed,  following  Painleve,  a  natural  family. 

Thus,  in  space  of  two  dimensions,  the  complete  system  of 
trajectories  for  a  given  conservative  field  of  force  consists  of  oo3 
curves  grouped  into  oo1  natural  families,  each  composed  of  oo2 
curves.  For  example,  in  the  case  of  ordinary  gravity  the  tra- 
jectories are  the  »3  vertical  parabolas  (in  a  given  vertical  plane), 
and  the  natural  families  are  formed  by  grouping  together  those 
parabolas  which  have  the  same  (horizontal)  line  as  directrix. 

In  space  of  three  dimensions,  the  complete  system  contains 
oo 5  trajectories  grouped  into  oo1  natural  families,  each  containing 
oo4  curves.  Examples  are  the  oo4  parabolas  with  vertical  axes 
whose  directrices  are  situated  in  a  fixed  horizontal  plane;  and 
the  oo4  circles  orthogonal  to  a  fixed  sphere.  The  simplest  ex- 
ample, corresponding  to  the  case  of  zero  force,  is  the  oo4  straight 

lines  of  space. 

34 


ASPECTS   OF   DYNAMICS.  35 

This  grouping  of  the  trajectories  according  to  the  values  of 
the  total  energy  constant,  that  is,  into  natural  families,  is  funda- 
mental in  most  dynamical  investigations  relating  to  conservative 
forces,  in  particular,  those  connected  with  the  principle  of  least 
action  and  the  developments  of  Hamilton  and  Jacobi.  From 
this  point  of  view,  dynamical  problems  relating  to  the  same 
field  of  force,  but  having  distinct  values  of  h,  are  considered  as 
essentially  distinct  problems.  Quoting  Darboux:  "This  re- 
striction is  in  accordance  with  the  spirit  of  modern  mechanics 
which  attaches  less  importance  to  force  than  to  energy,  and  which 
permits  us  to  regard  as  distinct  two  problems  in  which  the  force 
function  or  work  function  is  the  same,  but  the  total  energy  is 
different." 

It  therefore  seems  of  interest  to  work  out  the  purely  geometric 
properties  of  natural  families.  According  to  the  principle  of 
least  action,  such  a  family  is  made  up  of  the  extremals  defined 
by  the  variation  problem 

J  ^W  +  h  ds  =  minimum, 

that  is,  the  curves  which  cause  the  first  variation  of  the  integral 
to  vanish.     This  follows  from  the  fact  that  the  speed  v,  in  the 

action  integral  jvds,  is  determined  by  the  energy  equation 

#=  2(W  +  h). 

Abstractly,  a  natural  family  of  curves  may  be  defined  as  one 
which  can  be  regarded  as  the  totality  of  extremals  connected 
with  a  variation  problem  of  the  form 

J  Fds  =  minimum, 

where  F  is  any  point  function,  that  is,  any  function  of  x,  y,  z  in 
the  three-dimensional  case. 

Such  families  arise  not  only  in  the  discussion  of  trajectories, 
but  also,  for  example,  in  the  discussion  of  brachistochrones, 
catenaries,  optical  rays,  geodesies,  and  contact  transformations. 


36  THE   PRINCETON  COLLOQUIUM. 

The  brachistochrone  problem  for  a  conservative  field  with  any 
work  function  W  leads  to  the  integral 


/*-; 


ds 


+  h 


Thus  the  complete  system  of  brachistochrones  is  made  up  of 
oo1  natural  families,  one  for  each  value  of  h. 

When  a  homogeneous,  flexible,  inextensible  string  is  suspended 
in  the  conservative  field,  the  forms  of  equilibrium,  which  are 
termed  catenaries  in  the  general  sense  of  the  word,  are  obtained 
by  rendering  the  integral 

f(W+K)ds 

a  minimum.  Hence  here  also  we  have  oo1  natural  families,  one 
for  each  value  of  h.* 

Consider  an  isotropic  medium  in  which  the  index  of  refraction 
v  varies  arbitrarily  from  point  to  point.  The  paths  of  light  in 
such  a  medium,  according  to  Fermat's  principle  of  least  time, 
are  determined  by  minimizing  the  integral  J  vds  and  hence  form 
a  single  natural  family.  This  is  the  most  concrete  way  of  defining 
a  natural  family. 

The  connection  with  the  theory  of  geodesies  is  obvious. 
Thus  in  the  two-dimensional  case  the  geodesies  of  the  surface 
whose  squared  element  of  length  (first  fundamental  form)  is 
X(:r,  y}(dxi  +  dy~)  are  found  by  minimizing  the  integral  f  VX  ds, 
and  hence  the  representing  curves  in  the  x,  y  plane  constitute 
a  natural  family.  Hence  if  any  surface  is  represented  conformally 
on  a  plane,  the  geodesies  are  pictured  by  a  natural  family  of 
curves  in  that  plane.  The  extension  to  more  variables  is  evident : 


*  The  complete  systems  of  oo5  brachistochrones  and  oo5  catenaries  have 
geometric  properties  distinct  from  each  other  and  from  those  of  the  »s 
trajectories:  no  quintuply  infinite  system  of  curves  can  be  at  the  same  time 
the  system  of  trajectories  for  some  field  and  the  system  of  brachistochrones 
or  catenaries  in  either  the  same  or  a  different  field.  The  distinctive  properties 
for  an  arbitrary  field  are  given  in  §  107,  p.  94.  Cf.  §  103. 


ASPECTS   OF  DYNAMICS.  37 

any  natural  family  in  any  space  may  be  obtained  by  conformal 
representation  from  the  geodesies  of  some  other  space.* 

As  a  last  application  we  consider  the  transformations  which 
Sophus  Lie  has  termed  the  infinitesimal  contact  transformations 
of  mechanics.  In  the  plane  case,  such  a  transformation  is 
defined  by  a  characteristic  function  of  the  special  form 
12(ar,  */)(!  +  y'  )*  and  is  characterized  by  the  fact  that  the  lineal 
elements  at  each  point  are  converted  into  the  elements  of  a 
circle  about  that  point  as  center.  The  path  curves  of  every 
contact  transformation  of  this  category  form  a  natural  family. 

§§  29-31.     CHARACTERISTIC  PROPERTIES  A  AND  B 

29.  Osculating  Circles — Property  A. — We  now  consider  the 
general  geometric  properties  of  a  natural  family  in  ordinary 
space,  that  is,  the  totality  of  oo4  extremals  connected  with  an 
integral  of  the  form 

(1)  fF(z,  y,  z)  Vi  +  y*  +  2'2  dx. 

The  differential  equations  of  the  family  are  then  the  corresponding 
Euler-Langrange  equations 

2/'' =(£, -</'£*)  (1  +  /  + A 

z"=  (Lz  -  z'LJ(l  +  y'2  +  z'2), 
where 

L  =  log  F. 

Of  the  oo 4  curves  in  this  family  oo2  pass  through  any  given 
point  p,  one  in  each  direction.  Our  first  result  is: 

THEOREM  1:  The  oo4  curves  in  any  natural  family  have  this 
property:  the  circles  which  at  any  point  p  of  space  osculate  the  oc2 
curves  passing  through  that  point,  have  a  second  point  P  in  common 
and  thus  form  a  bundle. 

*  A  natural  family  on  a  given  surface  may  be  regarded  as  a  family  of 
pseudo-geodesies,  that  is,  one  which  may  be  obtained  as  the  conformal  picture 
of  the  geodesies  on  some  other  surface. 


38  THE   PRINCETON  COLLOQUIUM. 

This  property  we  shall  refer  to  as  property  A.  In  the  discussion 
it  is  convenient  to  decompose  it  into  these  two  statements,  also 
relating  to  the  oo2  curves  through  a  given  point: 

(A\)  The  osculating  planes  constructed  at  the  common  point 
form  a  pencil. 

(Az)  The  centers  of  curvature  lie  in  a  plane  perpendicular 
to  the  axis  of  the  pencil  of  osculating  planes. 

A  proof  of  the  theorem  stated  is  easily  obtained  by  regarding 
the  family  as  made  up  of  dynamical  trajectories.  Property  A\ 
results  from  the  fact  that  the  osculating  plane  of  a  trajectory 
always  passes  through  the  force  vector.  Property  A2  is  proved 
by  noting  that  those  trajectories  through  a  given  point,  which 
correspond  to  the  same  value  of  the  total  energy  h,  are  all 
described  with  the  same  initial  velocity  flo-  The  radius  of 
curvature  at  the  initial  point  is  given  by  the  formula 

r  =  r<?/N, 

where  AT  denotes  the  component  of  the  force  along  the  principal 
normal.  Since  AT  is  the  orthogonal  projection  of  a  fixed  vector, 
the  locus  of  its  terminal  point  will  be  a  sphere  through  the  initial 
point.  The  conclusion  then  follows  from  the  fact  that  r  varies 
inversely  as  N. 

The  following  analytical  discussion  has  the  advantage  of 
answering  the  converse  question  which  naturally  arises:  Are 
there  other  systems  with  property  At 

The  differential  equations  of  any  system  of  oo4  space  curves, 
one  determined  by  each  lineal  element  of  space,  may  be  assumed 
in  the  form 

(3)  y"  =  f(x,  y,  z,  y',  z'),       z"  =  g(x,  y,  z,  y',  *')• 

Property  A\  requires  that  at  each  point  there  shall  be  a  certain 
direction  through  which  all  the  osculating  planes  at  that  point 
must  pass.  Let  the  direction  in  question  be  given  by  the  ratios 
of  three  arbitrary  point  functions 

(4)  <t>(x,  y,  z),        t(r,  y,  z),        x(x,  y,  z) ; 


ASPECTS   OF   DYNAMICS.  39 

then  the  requisite  condition  is 

~v  £  =  X-g> 

</"    *  -  ytf 

Property  .da  requires  that  the  centers  of  curvature  shall  lie  in 
a  plane  perpendicular  'to  the  direction  (4) ;  hence 

(6)  4>X  +  *Y  +  XZ  =  1, 

where  X,  Y,  Z  denote  the  coordinates  of  the  center  relative 
to  axes  with  the  common  point  as  origin.  Using  the  general 
formulas  for  the  center  of  curvature,  and  combining  with  (5),  we 
find 

THEOREM  2 :  The  differential  equations  of  any  system  of  curves 
possessing  property  A  are  of  the  form 

y"=  (^- 

(7) 

2"=    (X- 

where  <£,  \J/,  x  are  arbitrary  functions  of  x,  y,  2.  The  converse  is 
valid  also. 

The  equations  (2)  are  seen  to  be  included  in  this  form,  hence 
the  result  certainly  holds  for  our  natural  systems,  as  stated  in 
theorem  1. 

30.  Hyper  osculations — Property  B. — The  circles  of  curvature 
at  a  given  point,  for  any  system  of  the  form  (7),  constitute  a 
bundle.  We  now  inquire  whether  any  of  these  circles  correspond 
to  four-point,  instead  of  three-point,  contact. 

If  a  twisted  curve  is  to  have  an  hyperosculating  circle  of  cur- 
vature at  a  given  point,  two  conditions  must  be  satisfied,  namely, 


(8) 


1     y'  z' 

0     y"  z" 

0     y'"  z'" 
dr 


=  0, 


40  THE  PRINCETON   COLLOQUIUM. 

The  first  of  these  states  that  the  osculating  plane  has  four-point 
contact  with  the  curve;  the  second,  in  which  r  denotes  the  radius 
of  curvature,  is  the  condition  for  the  existence  of  an  osculating 
helix,  i.  e.,  one  with  four-point  contact.  When  both  conditions 
hold  the  helix  is  simply  the  circle  of  curvature,  which  then  has 
hypercontact. 

Applying  these  conditions  to  the  curves  defined  by  (7),  we  find, 
from  (8), 

and,  from  (9), 

(11)     (1  +  yf  +  2'2)2</K/>'  -(</>  +  y't  +  z'x) 

,'    _|_    7yU'  _|_    R'Y'    1 

=  o, 


where  the  indicated  summations  extend  over  <£,  \(/,  x  and  where  </>', 
for  example,  denotes  02  +  y'$v  +  z'<f>z. 

Since  we  wish  to  discuss  the  oo2  curves  through  a  given  point, 
we  may  simplify  our  equations  considerably  by  taking  the  axis 
of  abscissas  in  the  special  direction  (4).  Then,  at  the  selected 
point,  \f/  and  x  vanish,  and  the  above  equations  reduce  to 

(10')  y'x'  -  z'f  =  0, 

(11')  (/  +  z'2)(4>'  -  02)  -  W  +  z'x'}  =  0. 

Neglecting  the  trivial  solutions  for  which  y'2  +  z'2  vanishes, 
we  may  reduce  this  pair  of  simultaneous  equations  to  the  form 


y  % 

This  set  of  equations  for  the  determination  of  y',  z'  is  of  a  familiar 
type,  namely,  that  arising  in  the  determination  of  the  fixed 
points  of  a  collineation,  and  is  easily  shown  to  admit  three  solu- 
tions.* Hence 


*  Of  course  in  special  cases  some  of  these  may  coincide,  or  the  number 
of  solutions  may  become  infinite.  The  theorem  stated  is  true  "  in  general  " 
in  so  far  as  it  omits  these  cases  which  are  definitely  assignable. 


ASPECTS   OF   DYNAMICS.  41 

THEOREM  3 :  The  curves  defined  by  equations  of  the  form  (7) 
are  such  that  through  each  point  there  pass  three  with  hyperosculating 
circles  at  that  point. 

Since  the  form  (7)  is  characterized  by  property  A,  it  follows 
that  the  existence  of  three  hyperosculating  circles  in  each  bundle 
is  a  consequence  of  property  A. 

We  state  two  further  properties,  found  by  considering  the 
conditions  (10')  and  (11')  separately. 

The  tangents  to  those  curves  of  a  system  (7)  which  pass  through  a 
given  point  and  there  have  an  hyperosculating  plane  form  a  quadric 
cone.  This  cone  passes  through  the  special  direction  (4). 

The  tangents  to  those  curves  which  have  an  osculating  helix  at  the 
given  point  form  a  cubic  cone.  This  cone  passes  through  the 
special  direction  (4)  and  through  the  minimal  directions  in  the 
plane  normal  to  that  direction. 

These  properties  hold  for  natural  families  since  they  hold  for 
all  systems  with  property  A.  By  comparing  (7)  with  (2),  we 
see  that  the  functions  0,  \f/,  x  in  the  case  of  a  natural  family  are 

(13)  0  =  Lx,        \f/  =  Lv,        x  =  L2; 
and  hence  are  connected  by  the  relations 

(14)  ^  -  xv  =  0,         x*-<t>*  =  0,        4>y-tx  =  0. 

We  now  inquire  what  is  the  effect  of  these  relations  on  the 

directions  of    the    hyperosculating    circles.     Introducing,    for 
symmetry, 

(15)  X  :  Y  :  Z  =  1  :  y'  :  z', 

we  may  write  our  equations  (12)  in  the  homogeneous  form 

(16)  -  xxX2+<t>2Z2+<f>yYZ-  xvXY+  (0*-02-  Xz)XZ= 0, 

In  virtue  of  (14),  each  of  the  quadric  cones  (16)  is  seen*  to  be 

*  The  condition  for  such  a  cone  is  that  the  sum  of  the  coefficients  of  X2, 
Y2,  and  Z2  shall  vanish. 


42  THE   PRINCETON  COLLOQUIUM. 

of  the  rectangular  type.  Hence  the  three  generators  common 
to  the  cones  must  be  mutually  orthogonal.  This  gives 

THEOREM  4  :  In  the  case  of  any  natural  family  the  three  hyper- 
osculating  circles  which  exist  in  any  bundle  are  mutually  orthogonal. 

We  refer  to  this  property  as  property  B. 

31.  The  relations  (14)  are  seen  to   be  necessary  as  well   as 
sufficient  for  the  orthogonality  in  question.     Hence  property  B 
is  the  equivalent  of  (14),  and  serves  to  single  out  the  natural 
families  from  the  more  general  class  defined  by  equations  of 
form  (7).     The  latter  form  was  characterized  by  property  A; 
hence  we  have  our 

FUNDAMENTAL  THEOREM:  A  system  of  oo4  curves,  one  for  each 
direction  at  each  point  of  space,  will  constitute  a  natural  family 
when,  and  only  when,  it  possesses  properties  A  and  B:  that  is,  the 
osculating  circles  at  any  given  point  must  form  a  bundle,  and  the 
three  hyperosculating  circles  contained  in  such  a  bundle  must  be 
mutually  orthogonal. 

§  32.    GENERAL  VELOCITY  SYSTEMS 

32.  The  most  general  system  with  property  A  is  represented 
by  differential  equations  of  the  form 

y/2  +  A 


and  thus  involves  three  arbitrary  functions.  Only  in  the  case 
where  these  functions  are  the  partial  derivatives  of  the  same 
function  is  the  system  a  natural  one.  We  now  point  out  a 
dynamical  problem  that  leads  to  the  general  type  (7):  this 
justifies  the  term  velocity  system  which  we  hereafter  employ  to 
denote  any  system  of  this  type. 

Consider  a  particle  (of  unit  mass)  moving  in  any  field  of  force, 
the  components  of  the  force  being  6,  \{/,  x-  The  equations  of 
motion  are  then 

x  =  <f>(x,  y,  z),         y  =  \l/(x,  y,  z),         z  =  x(x,  y,  z}. 
If  the  initial  position  and  the  initial  velocity  are  given  the  motion 


ASPECTS   OF   DYNAMICS.  43 

is  determined.  If  only  the  initial  position  and  direction  of 
motion  are  given,  the  osculating  plane  will  be  determined  but 
the  radius  of  curvature  r  will  depend  for  its  value  on  the  initial 
speed  v.  Hence,  in  addition  to  the  usual  formula 


there  must  be  a  formula  expressing  v2  in  terms  of  x,  y,  z,  y',  z',  r. 
This  is  furnished  by  the  familiar  equation 

v2  =  rN, 

where  N  denotes  the  (principal)  normal  component  of  the  force, 
so  that 

A72          ^2_L      /2_l         2  (4>  +  tfty  +  *'xf 

=  *-+*2+x2-  ' 


The  result  may  be  written  in  the  two  (equivalent)  forms 

2  _    Qfr-yfrXl  +  y'S+Z*)   _    (X-2»(l  +  y'2+2'2) 

"  y"  z" 

In  the  actual  trajectory  v  varies  from  point  to  point.  If  now  we 
replace  v2  in  this  result  by  some  constant,  say  l/c,  the  resulting 
equations  may  be  written 


The  curves  satisfying  these  differential  equations  —  they  are  not 
in  general  trajectories  —  we  define  as  velocity  curves.  For  any  field 
a  curve  is  a  velocity  curve  corresponding  to  the  speed  v0,  provided 
a  particle  starting  from  any  lineal  element  of  the  curve  with 
that  speed  describes  a  trajectory  osculating  the  curve.  In  a 
given  field  of  force  there  are  oo5  trajectories  and  oo5  velocity 
curves.*  If  c  is  given  we  have  oo4  velocity  curves.  In  particular 


*  The  properties  of  a  complete  system  of  oo5  velocity  curves  are  analogous 
to,  but  distinct  from,  those  of  a  complete  system  of  trajectories.     Cf.  p.  94. 


44  THE  PRINCETON   COLLOQUIUM. 

if  c  (and  hence  v)  is  taken  to  be  unity,  our  equations  become 
precisely  (7). 

Any  system  of  oo4  curves  possessing  property  A,  that  is,  any 
system  (7),  may  be  regarded  as  the  totality  of  velocity  curves  cor- 
responding to  unit  velocity  in  some  (uniquely  defined]  field  of  force. 

Only  when  the  field  is  conservative  do  the  velocity  systems  for 
each  value  of  v  (or  c)  become  natural  systems.  The  trajectories 
also  are  in  this  case  made  up  of  oo1  natural  families,  one  for  each 
value  of  the  energy  constant  h;  but  the  two  sets  of  natural  families 
are  distinct.  The  determination  of  a  velocity  system  in  one 
conservative  field  is  equivalent  to  the  determination  of  a  tra- 
jectory system  in  another  conservative  field,  and  vice  versa. 
We  find  in  fact  the  following  explicit  result: 

//  two  conservative  fields  with  work  functions  W\  and  JF2  satisfy 
the  relation* 

"iWi 

Wz  =  ae^l  -  h, 

then  the  oo4  velocity  curves  for  the  speed  VQ  in  the  first  field  coincide 
with  the  oo4  trajectories  for  the  constant  of  energy  h  in  the  second 
field.^ 

§  33.  RECIPROCAL  SYSTEMS 
33.  With  any  velocity  system  S 


y= 

there  is  connected  a  definite  point  transformation  T:  for  in  virtue 
of  property  A  to  any  point  p  corresponds  a  definite  point  P, 
the  osculating  circles  constructed  at  the  first  point  all  passing 
through  the  second  point.  The  transformation  T  is  explicitly 


(T) 


*  We  note  that  if  W\  is  left  unaltered  and  rc  varied,  TFj  takes  quite  distinct 
forms.  The  oo1  velocity  systems  in  a  given  field  do  not  constitute  the  com- 
plete system  of  oc6  trajectories  in  any  field  whatever. 

t  It  is  seen  that  the  two  fields  have  the  same  equipotential  surfaces  and 
therefore  the  same  lines  of  force.  (Central  fields  therefore  correspond  to 
central  fields.) 


ASPECTS   OF  DYNAMICS.  45 

It  is  thus  entirely  general.  To  an  arbitrary  transformation* 
corresponds  a  definite  velocity  system.  In  particular,  to  the 
inverse  transformation  T~l  there  corresponds  a  certain  system 
S',  which  we  define  as  reciprocal  to  S. 

Hence  to  a  general^  velocity  system  S,  that  is,  any  system  possessing 
property  A,  there  corresponds  a  definite  reciprocal  velocity  system 
S'.  The  osculating  circles  of  those  curves  of  system  S  which  pass 
through  any  point  p  are  at  the  corresponding  point  P  the  osculating 
circles  of  the  curves  of  the  system  S'  passing  through  P. 

Consider  the  bundle  of  circles  determined  by  two  corresponding 
points  p  and  P.  We  know  that  three  of  these  circles  have 
hypercontact  with  ^-curves  at  p,  and  three  have  hypercontact 
with  <S'-curves  at  P.  It  is  not  obvious  that  the  circles  so  ob- 
tained really  coincide.  Omitting  the  rather  long  proof,  we 
merely  state  the  result. 

Reciprocal  velocity  systems  have  the  same  hyperosculating  circles: 
the  three  circles  hyperosculating  curves  of  the  given  system  S  at 
any  point  p  also  hyperosculate  curves  of  the  reciprocal  system  Sf 
at  the  corresponding  point  P. 

It  follows  at  once  that  if  S  possesses  property  B  (that  is 
mutually  orthogonal  hyperosculating  circles)  the  same  will  be 
true  of  S'.  This  means  that  whenever  system  S  is  natural  so  is  S'. 

The  reciprocal  of  a  natural  family  is  always  a  natural  family. 

We  may  restate  this  in  optical  terms  as  follows:  With  any 
isotropic  medium,  defined  by  its  index  of  refraction  v(x,  y,  z), 
there  is  connected  a  certain  reciprocal  medium  with  an  index  of 
refraction  v(x,  y,  z):  the  rays  of  light  in  this  second  medium, 
namely,  the  extremals  of 

j  v(x,  y,  z)ds  =  minimum, 

form  the  system  reciprocal  to  that  formed  by  the  rays  of  light 

*  It  may  even  degenerate  but  must  not  be  merely  the  identical  trans- 
formation. We  however  exclude  systems  with  degenerate  T"s  from  the  rest 
of  the  discussion:  we  assume  that  the  jacobian  does  not  vanish,  so  that  the 
inverse  transformation  exists. 

t  See  preceding  footnote. 


46  THE   PRINCETON   COLLOQUIUM. 

in  the  given  medium,  namely,  the  extremals  of 

j  v(.r,  y,  z)ds  =  minimum. 

The  actual  calculation  of  v  from  v  requires  only  operations  that 
are  performable  in  the  Lie  sense,  namely,  eliminations  and  dif- 
ferentiations. See  Transactions  of  the  American  Mathematical 
Society,  volume  10  (1909),  page  213. 

§  34.  CHARACTER  OF  THE  TRANSFORMATION  T 

34.  The  transformation  T  (from  point  p  to  point  P)  associated 
with  the  most  general  system  possessing  property  A  is,  as  we  have 
seen,  entirely  arbitrary.  The  question  arises  what  is  the  pecu- 
liarity of  T  if  the  given  system  is  of  the  natural  type.  The 
answer  to  this  will  furnish  an  equivalent  of  property  B,  and 
will  thus  make  it  possible  to  characterize  natural  families  with- 
out introducing  hyperosculating  circles. 

The  problem  is  to  describe  geometrically  the  class  of  trans- 
formations of  the  form 

Y=    i          2Lx  y=  2L» 

*"  *  ' 


= 

*  '          y~  LJ+LJ+L*  ' 

2L, 


^ 


depending  on  one  arbitrary  function  L  of  x,  y,  z,  instead  of  three 
independent  functions  required  in  a  general  point  transformation, 

X  =  &(x,  y,  2),         Y  =  V(x,  y,  2),        Z  =  X(*,  y,  2). 

For  a  general  (analytic)  point  transformation  the  bundle  of 
lineal  elements  at  any  point  is  converted  linearly  into  the  bundle 
at  the  corresponding  point.  Are  there  any  elements  which  go 
over  into  parallel  elements?  It  is  well  known  that  there  are 
three.  If  in  particular  these  three  elements  are  mutually  per- 
pendicular (for  every  point  of  space),  we  obtain  a  certain  category 
of  transformations  wrhich  mav  be  termed  Darboux*  transforma- 


*  See  Proceedings  of  the  London  Mathematical  Society,  1900. 


ASPECTS   OF  DYNAMICS.  47 

tions  or  deformations.     They  are  analytically  of  the  form 
X  =  /z,         Y  =  fv,        Z  =  fz, 

involving  one  arbitrary  function.  Obviously  this  is  not  the  class 
we  desire. 

We  next  ask  whether  in  the  general  transformation  there  are 
any  elements  at  a  given  point  p  each  of  which  is  turned  into  a 
cocircular  element  at  the  corresponding  point  P.  This  is,  in  a 
way,  a  case  correlative  to  the  Darboux  case:  for  whether  two 
elements  in  space  are  parallel  or  cocircular  they  have  in  common 
the  properties  that  they  are  coplanar  and  equally  inclined  to  the 
line  pP  joining  their  points.  It  is  found  that  there  are  always 
three  such  elements  at  any  point.  If  we  require  these  to  be 
mutually  orthogonal,  we  obtain  precisely  the  transformations 
connected  with  natural  families. 

A  system  of  oo4  space  curves  possessing  property  A  will  form  a 
natural  family  when  and  only  when  the  associated  transformation  T 
(from  point  p  to  point  P)  has  the  following  property:  the  three  lineal 
elements  at  p  each  of  which  is  converted  into  a  cocircular  element 
at  P  are  mutually  orthogonal. 

We  have  thus  obtained  an  equivalent  for  property  B.  It 
may  be  shown  synthetically  that  the  three  directions  just  de- 
scribed (cocircular  elements)  always  coincide  with  the  directions 
of  the  hyperosculating  circles.  The  orthogonality  of  the  one 
triple  amounts  to  the  same  thing  as  the  orthogonality  of  the 
other. 

It  may  be  remarked  that  the  class  of  transformations  connected 
with  all  natural  systems  do  not  form  a  group.  It  is  obvious  how- 
ever that  the  inverse  of  any  member  of  the  class  is  contained  in 
that  class.  This  is  the  essence  of  the  law  of  reciprocity  for  natural 
systems,  previously  obtained  by  a  different  method. 

§§  35-44.    THE  CONVERSE  OF  THOMSON  AND  TAIT'S  THEOREM 

35.  It  is  well  known  that  if  straight  lines  are  drawn  orthogonal 
to  any  given  surface  they  will  necessarily  be  orthogonal  to  an 


48  THE   PRINCETON   COLLOQUIUM. 

infinitude  of  surfaces  (namely  the  surfaces  parallel  to  the  given 
surface).  Thomson  and  Tait  in  their  Natural  Philosophy  showed 
that  this  property  of  the  <»4  straight  lines  of  space  holds  for  the 
oo4  trajectories  described  in  any  conservative  field  with  the  same 
total  energy,  that  is,  for  any  natural  family.  The  writer  has 
proved  that  no  other  families  of  curves  have  the  property:  it  is 
entirely  characteristic  of  the  natural  type.*  We  first  state  the 
original  theorem  in  connection  with  the  general  theory  of  the 
calculus  of  variations,  and  then  take  up  the  converse  theorem. 
Later  a  second  converse  question  is  discussed. 

35'.  Thomson  and  Tail's  Theorem. — We  ha"ve  seen  that  a 
natural  family  of  curves  in  space  may  be  regarded  as  the  totality 
of  extremals  of  a  variation  problem  of  the  particular  form 

(1)  J  =  fF(x,  y,  z)ds, 

where  F  is  a  point  function,  ds  is  the  element  of  length 


and  the  integral  is  taken  between  fixed  end  points. 

It  is  easily  shown  that  for  integrals  of  this  form,f  and  for  no 
others,  the  relation  of  transversality ,  in  the  sense  of  the  calculus 
of  variations,  amounts  merely  to  orthogonality.  This  suffices 
to  distinguish  our  type  among  variation  problems  of  the  general 
form 
(2)  //(*,  y,  z,  y'  z'}dx. 

But  of  course  it  does  not  serve  as  a  complete  geometric  test  for 
a  natural  family.  What  is  the  geometric  character  of  the  systems 
of  oo4  extremals  connected  with  any  variation  problem(2)? 
This  is  an  unsolved  question  in  the  calculus  of  variations. % 

*  At  least  in  the  case  of  space  of  three  dimensions.  Cf.  Trans.  Amer. 
Math.  Soc.,  vol.  11  (1910),  pp.  121-140. 

t  Cf .  Bolza,  Variationsrechnung,  p.  691 ;  also  p.  146  for  the  two-dimensional 
problem  due  to  Hedrick. 

t  See  the  author's  paper,  "Systems  of  extremals  in  the  calculus  of  variations," 
Butt.  Amer.  Math.  Soc.,  vol.  13  (1908),  pp.  289-292. 


ASPECTS   OF   DYNAMICS.  49 

We  are  concerned  here  only  with  the  integrals  of  special  form  J, 
defining  natural  families.  Applying  Kneser's  fundamental 
theorem  on  transversals,*  we  have  this  well-known  result:  If  from 
the  points  of  any  surface  2  we  construct  the  extremals  orthogonal 
to  the  surface,  and  on  each  lay  off  an  arc  so  that  the  integral  J 
takes  some  constant  value,  then  the  locus  of  the  end  points  is  a 
surface  which  is  also  orthogonal  to  the  extremals. 

36.  This  is  known  as  the  theorem  of  Thomson  and  Tait.  It 
was  obtained  by  them  in  connection  with  the  dynamics  of  a 
particle  moving  in  a  conservative  field  —  the  first  interpretation 
of  a  natural  family  considered  in  §  28.  Here  F(x,  y,  z)  represents 
the  speed  v,  as  determined  by  the  energy  equation 

tf=  2(W+h), 

where  W  denotes  the  work  function  (negative  potential),  and 
the  mass  is  assumed  to  be  unity.  Of  course  h  has  a  fixed  value. 
We  quote  the  original  statement  of  the  theorem: 

"  If  from  all  points  of  an  arbitrary  surface  particles  not  mu- 
tually influencing  one  another  be  projected  normally  with  the 
proper  velocities  [so  as  to  make  the  sum  of  the  kinetic  and  potential 
energies  have  a  given  value];  particles  which  they  reach  with 
equal  actions  lie  on  a  surface  cutting  the  paths  at  right  angles." 

The  integral  J,  in  this  case,  represents  the  action 


The  oo  l  surfaces  cutting  the  curves  orthogonally  thus  appear  as 
surfaces  of  equal  action. 

The  corresponding  statement  for  brachistochrones  is  sometimes 
called  the  theorem  of  Bertrand:]  From  the  points  of  any  surface 
draw  the  brachistochrones  normal  to  the  surface  and  on  each  lay 
off  lengths  so  that  the  time  of  transit  is  equal  to  a  given  quantity; 
then  the  locus  of  the  end  points  will  be  another  surface  orthogonal 

*  Bolza,  pp.  131  and  691. 

t  Cf.  Routh,  Dynamics  of  a  Particle  (1898),  p.  376.     According  to  Appell, 
Mecanique  rationelle,  vol.  1  (1909),  p.  466,  this  result  was  indicated  by  Euler. 
12 


50  THE   PRINCETON    COLLOQUIUM. 

to  the  brachistochrones.     Here  the  integral  J  represents  the  time 

ds 


h) ' 

so  that  the  orthogonal  surfaces  appear  as  surfaces  of  equal  time. 
Corresponding  statements  may  be  made,  of  course,  for  the  other 
interpretations  leading  to  natural  families.  The  most  concrete 
aspect  is  obtained  by  using  the  language  of  optics.  Here  the 
integrand  function  is  simply  the  index  of  refraction  v(x,  y,  z), 
varying  from  point  to  point  in  any  (isotropic)  medium,  and  the 
integral  J  vds  is  proportional  to  the  time.  The  paths  of  light  in 
such  a  medium  form  a  (single)  natural  family,  and  every  natural 
family  may  be  obtained  in  this  way.  The  <x>2  rays  (in  general 
curved)  starting  out  normally  from  anys  urface  admit  oo1 
orthogonal  surfaces.  These  present  themselves  as  surfaces  of 
equal  time.  We  shall  describe  them  as  a  set  of  wave  fronts  or 
wave  surfaces. 

37.  The  geometric  part  of  the  theorem  of  Thomson  and   Tail 
may  be  stated  as  follows :  In  any  natural  family  of  oo 4  space  curves, 
the  oo 2  curves  which  meet  any  surface  orthogonally  always  form  a 
normal  congruence. 

Is  this  geometric  property,  which  we  shall  refer  to  as  the 
Thomson-Tait  property,  characteristic?  This  is  in  fact  the  case. 
We  shall  prove,  namely,  the  following 

CONVERSE  THEOREM.  //  a  quadruply  infinite  system  of  curves  in 
space  is  such  that  oo2  of  the  curves  meet  an  arbitrarily  given  surface 
orthogonally*  and  always  form  a  normal  congruence  (that  is,  admit 
an  infinitude  of  orthogonal  surf  aces) ,  then  the  system  is  of  the  natural 
type,  that  is,  it  may  be  identified  with  the  extremal  system  belonging 
to  an  integral  of  the  form  jF(x,  y,  z)ds. 

38.  The  result  is  simple  but  the  proof  is  rather  long.     We  give 
the  essential  steps. 

Consider  an  arbitrary  quadruply  infinite  system  of  curves  in 

*  This  means  the  same  as  requiring  that  one  curve  of  the  system  passes 
through  each  point  of  space  in  each  direction. 


ASPECTS   OF  DYNAMICS.  51 

space,  assuming  that  one  passes  through  each  point  in  each 
direction.  Such  a  system  may  be  defined  by  a  pair  of  differential 
equations  of  the  second  order 

(1)          y"  =  F(x,  y,  z,  y',  2'),        z"  =  G(x,  y,  z,  y',  z'), 

where  F  and  G  are  uniform  functions  which  we  assume  to  be 
analytic  in  the  five  arguments.  Denoting  the  initial  values  of 
x,  y,  z,  y',  z',  which  may  be  taken  at  random,  by  x,  y,  z,  p,  q 
respectively,  and  j  employing  X,  Y,  Z  as  current  coordinates, 
we  may  write  the  solutions  of  (1)  in  the  form 

7  =  y  +  p(X  -  x} 
(2) 

Z=  z+q(X-x)  + 

Here  F  and  G  are  expressed  as  functions  of  x,  y,  z,  p,  q',  and  M 
and  N,  found  by  differentiating  (1),  are  given  by 

M  =  FX  +  PFy  +  qF,  +  FFP  +  GFq, 
N  =  Gx  +  pGy  +  qGz  +  FGP  +  GGq. 

The  terms  of  higher  order  will  not  be  needed  in  our  discussion. 
Equations  (2)  involve  five  arbitrary  parameters  but  of  course 
represent  only  oo4  curves. 

Consider  now  an  arbitrary  surface  2 

(4)  z=/(*,y). 

At  each  point  of  this  surface  and  normal  to  it  a  definite  curve 
of  the  given  family  (1)  may  be  constructed.  A  certain  congruence 
will  thus  bt  determined.  We  wish  to  express  the  condition 
that  this  shall  be  of  the  normal  type,  that  is,  that  the  oo2  curves 
shall  admit  a  family  of  orthogonal  surfaces. 

The  direction  normal  to  the  surface  S  at  any  point  is  given  by 

1  -P  -q  =  fx  :/„  :  -  1, 
so  that 

(5)  p  =  P(x,  y)}        q  =  Q(x,  y), 
where 

(50  P  =/«//*,      Q--I//X. 


52  THE   PRINCETON  COLLOQUIUM. 

These  functions  are  connected  by  the  relation 
(5")  PQX  -  q?x  -  Qv  m  0. 

The  equations  of  the  »2  curves  corresponding  to  the  given 
initial  conditions  may  now  be  written 

X  =  z+t, 

(6)  Y  =  y  +  Pt  +  IF?  + 


where  t  takes  the  place  of  A'  —  x  in  (2),  and  where  the  bars 
indicate  that  the  substitution  (4),  (5)  has  been  carried  out,  so 
that,  for  example, 

(7)  F(x,y]  =F(x,y,f,P,Q). 

The  coefficients  of  the  powers  of  t  in  (6)  are  thus  functions  of  the 
two  parameters  x,  y. 

The  general  condition  for  a  normal  congruence  given  in  para- 
metric form  is* 

(8)  (Y'XY)  -  (Z'ZX)  +  Y'(Z'YZ)  -  Z'(Y'YZ}  =  0, 

where  the  parentheses  denote  jacobians  taken  with  respect  to  t, 
x,  y,  and   Y',  Z'  denote   the  derivatives   of  Y,  Z  respectively 
with  respect  to  t. 
Expanding  (8)  in  powers  of  t  in  the  form 

(9)  a0+  fli<  +  W-  +  •••, 

we  find  that  00  vanishes  in  consequence  of  (5")-  This  is  as 
it  should  be,  since  our  oo2  curves  are  orthogonal  to  S  by  con- 
struction. 

The  terms  containing  the  first  power  of  t  give 


*  We  may  also  use  the  convenient  form  due  to  Beltrami.     Cf.  Bianchi- 
Lukat,  Differentialgeometrie,  p.  340. 


ASPECTS   OF   DYNAMICS.  53 


From  (6')  we  find 

Fx  =  Fx  +  FJX  +  FPPX  +  FqQx, 


with  corresponding  results  for  Gx  and  Gv.     Substituting  these 
values,  and  observing  from  (5)  and  (5')  that 


/*  =  -  l/<?,      /»  =  -  P/Q,      Qy  =  P<3*  -  QPX, 

we  may  reduce  (10)  to 


(10')        X  {QFx-Fz-PGx+Gy+(QFp-QGq-PGp)Px 

+GpPv+QFqQx}=0. 

This  is  then  a  necessary  condition  in  order  that  the  oo2  curves 
belonging  to  the  quadruply  infinite  system  (1)  and  orthogonal 
to  the  surface  (4)  shall  form  a  normal  congruence.  The  result 
is  to  hold  in  virtue  of  (4)  and  (5). 

It  is  of  course  not  a  sufficient  condition.  It  merely  expresses 
the  fact  that  the  curves  orthogonal  to  S  are  also  orthogonal  to 
some  consecutive  surface,  that  is,  that  the  congruence  is  approxi- 
mately normal  to  the  first  degree. 

Our  main  problem  is  to  find  all  systems  (1)  which  have  the 
orthogonality  property  with  respect  to  every  base  surface  S. 
It  is  then  necessary  that  (10')  should  be  true  for  an  arbitrary 
function  f(x,  y).  The  function  can  be  so  selected  that  for  any 
chosen  values  of  x  and  y  the  quantities  /,  P,  Q,  Px,  Py,  Qx, 
shall  take  arbitrary  numerical  values;  for  the  only  relation  to 
be  fulfilled  is  (5")  and  this  merely  determines  Qv.  The  con- 
dition (10')  must  therefore  hold  identically.  Arranging  it  in  the 
form 
(10")  (1  +  P2  +  Q2)C0  +  QdQ,  +  C,Py  -  C,PX  =  0, 

and  equating  coefficients  to  zero,  we  find 

Co  =  qFx  -F.-  PGX  +  Gy  =  0, 

d  =  (1  +  p2  +  q*}Fq  -  2qF  =  0, 
(11)  C2  =  (1  +  p2  +  <?)GP  -  2pG  =  0, 


2pqF  -  2(p*  +  q*)G  =  0. 


54  THE   PRINCETON  COLLOQUIUM. 

Integration  of  the  second  and  third  of  these  partial  differential 
equations  gives 

F  =  fi(p,  *,  y,  «)U  +  P2  +  <72),        G  =  gi(q,  x,  y,  z)(l  +  f  +  g2), 

where  f\  and  g\  denote  unknown  functions  of  the  four  arguments 
indicated.  Substituting  these  values  in  the  fourth  equation, 
we  find  /ip  =  g\q,  and  therefore 

fi  =  t  -  p<t>,        g\  =  X  -  q<t>, 

where  0,  \f/,  x  are  functions  of  x,  y,  z  only.  The  general  solution 
of  the  last  three  equations  of  the  set  (11)  is  therefore 

(12)  F=(^-^)(l+p2+g2),        G=(x-<?</>)(l+p2+72). 

We  have  still  to  satisfy  the  first  equation  of  (11),  which  now 
reduces  to 

(13)  *.  -  x,  +  p(x*  -  0.)  +  q(4>y  -  U  =  0. 

The  functions  <f>,  \f/,  x  must  therefore  satisfy  the  equations 

(13')     ,k  -  Xv  =  o,      x*  -  <i>*  =  o,      «/>,  -  ^  =  o, 

and  hence  are  expressible  as  the  derivatives  of  a  single  function 

in  the  form 

(13")  0  =  Lx,       t=Ly,        x  =  Lz. 

The  solutions  of  the  set  (11)  are  therefore 

F=  (Ly-pLx)(l  +  p*  +  q*), 
G=  (L2-PLz)(l  +  p2  +  92), 

involving  an  arbitrary  function  L  of  x,  y,  z.  The  resulting  system 
(1)  is  thus  recognized  to  be  a  natural  family.  This  gives  our 
fundamental  converse  theorem. 

39.  In  the  above  discussion  use  has  been  made,  not  of  the 
complete  condition  for  a  normal  congruence,  but  only  of  con- 
dition (10')  derived  from  the  terms  of  the  first  order  in  t.  We 
may  therefore  state  a  stronger  converse  result  as  follows: 


ASPECTS   OF  DYNAMICS.  55 

The  only  systems  of  oo4  curves  which  have  the  property  that 
the  curves  orthogonal  to  any  surface  are  always  orthogonal  to  some 
infinitesimally  adjacent  surface  are  those  of  the  natural  type. 

If  a  congruence  of  curves  meets  two  neighboring  surfaces 
orthogonally  it  need  not  meet  oo1  surfaces  orthogonally,  and 
therefore  it  approximates  to,  but  need  not  coincide  with,  a  normal 
congruence.  The  above  theorem  shows  however  that  if  the 
weak  requirement  of  approximate  normal  character  be  imposed 
on  all  the  congruences  obtained  from  the  given  quadruply  infinite 
system,  they  will  all  be  exactly  normal. 

40.  We  may  further  strengthen  our  theorem  by  demanding  the 
orthogonality  property  for  some  instead  of  all  surfaces.  Our 
fundamental  equations  (11)  resulted  from  the  fact  that  x,  y,  z, 
f,  P,  Q,  Px,  Py,  Qx  might  receive  arbitrary  numerical  values. 
It  will  therefore  be  sufficient  to  take  a  manifold  of  surfaces 
sufficiently  large  to  leave  these  quantities,  or  the  equivalent 
quantities 

X,  y,          Zy          Zx,  Zy,          Zxx>          Zj;y,  Zyy, 


unrestricted.  Since  these  quantities  define  a  differential  surface 
element  of  the  second  order,  we  may  state  the  result  as  follows: 

The  converse  theorem  remains  valid  if,  instead  of  considering 
all  base  surfaces,  we  employ  a  manifold  of  surfaces  sufficiently 
large  to  include  all  the  oo8  possible  differential  elements  of  the 
second  order. 

41.  The  Thomson-Tait  theorem  holds  of  course  even  when  the 
base  S  shrinks  to  a  curve  or  a  point:  there  will  still  be  a  normal 
congruence  orthogonal  to  the  curve  or  point  (in  the  latter  case 
orthogonality  means  simply  passage  through  the  point).  We 
state  a  number  of  results  obtained  in  this  connection. 

If  for  an  arbitrary  curve  as  base  the  corresponding  oo2  orthog- 
onal curves  of  a  given  quadruply  infinite  system  always  form 
a  normal  congruence,  the  given  system  is  necessarily  natural. 

If  we  require  each  of  the  congruences  here  considered  to  be 
of  approximately  normal  character,  a  more  general  type  of  system 


56  THE   PRINCETON   COLLOQUIUM. 

is  obtained,  namely  the  velocity  type  of  §  32.  The  velocity  type 
is  thus  characterized  by  the  fact  that  those  curves  of  the  system 
which  meet  an  arbitrary  curve  orthogonally  are  orthogonal  to 
some  infinitesimally  adjacent  (of  course  tubular)  surface.  We 
may  even  restrict  ourselves  to  the  case  where  the  base  is  a  curve 
of  the  given  system,  or  the  case  where  it  is  any  straight  line. 

42.  Suppose  next  that  the  base  is  an  arbitrary  point.     Are 
natural  families  the  only  families  of  co4  curves  such  that  the  oo2 
curves  passing  through   any  point  form  a  normal  congruence? 
A  discussion  shows  that  this  is  not  the  case.     There  exist  families 
not  of  the  natural  type,  for  example,  that  defined  by  the  dif- 
ferential equations 

y"  =  y'\      *"  =  o, 

with  the  restricted  property  stated.  To  find  all  such  systems 
would  be  a  rather  difficult,  but  certainly  an  interesting,  under- 
taking. The  result  would  of  course  include  the  natural  type  as 
a  special  case. 

43.  It  will  not  however  be  the  velocity  type.      It  may  be 
shown  in  fact  that  the  only  velocity  systems  for  which  the  curves 
passing  through  an  arbitrary  point  constitute  always  a  normal 
congruence  are  those  of  the  natural  type.     Recalling  the  fact 
that  the  velocity  type  is  characterized  by  property  A,  we  may 
give  a  new  characterization  of  the  natural  type  as  follows: 

Natural  families  are  the  only  quadruply  infinite  systems  of  curves 
in  space  such  that  the  oo2  curves  through  an  arbitrary  point  admit 
an  infinitude  of  orthogonal  surfaces,  and  such  that  the  osculating 
circles  constructed  at  the  common  point  form  a  bundle. 

44.  It  may  also  be  shown  that  if  for  every  point  and  every 
straight  line  as  base  the  corresponding  congruence  is  normal, 
the  system  will  be  natural.     To  have  a  velocity  system  it  is 
sufficient  to  demand  that  the  congruence  corresponding  to  an 
arbitrary  straight  line  shall  be  approximately  normal.     To  have 
a  natural  system  it  is  sufficient  to  demand  approximate  normality 
for  the  congruences  corresponding  to  arbitrary  straight  lines  and 
plane?. 


ASPECTS  OF  DYNAMICS.  57 

§§  45-53.    WAVE    PROPAGATION    IN    AN    ISOTROPIC    MEDIUM  : 
PROPERTIES  OF  WAVE  SETS 

45.  The  optical  interpretation  of  a  natural  family  and  the 
Thomson-Tait  property  suggest  certain  sets  of  surfaces  which 
we  shall  now  study. 

Consider  a  given  medium  defined  by  its  index  of  refraction 
v(x,  y,  z)  given  as  a  function  of  position.  The  rays  (in  general 
curved  lines)  are  the  oo4  extremals  of 

(1)  Jv(x>  y>  2)^*  =  minimum; 

they  form  the  natural  family,  whose  differential  equations  are 

</"  =  (Lv-y'Lx-)(l  +  y'*  +  z'\ 

«"-  (i.-a'i,)(l  +  y'2  + A 
where 
(2')  L  -  log  v. 

The  oo 2  rays  orthogonal  to  any  selected  surface  S  form  a 
normal  congruence,  that  is,  are  orthogonal  to  a  set  of  oo1  surfaces. 
A  disturbance  originating  in  the  medium  on  the  surface  S  will 
be  propagated  in  the  medium  through  this  set  of  surfaces,  which 
we  term  a  set  of  wave  fronts.  In  the  given  medium  an  arbitrary 
surface  belongs  to  one  and  only  one  of  these  wave  sets.  A  single 
surface  is  thus  of  arbitrary  character,  but  the  sets  of  surfaces 

(3)  f(x,  y,  z)  =  constant 

that  may  be  wave  sets  are  restricted  by  the  Hamilton- Jacob i 
equation 

(4)  /,* +/„»+/.»=  v\ 

The  given  medium  defines  also  a  certain  set  of  level  surfaces 
v(x,  y,  z)  =  constant. 

This,  it  should  be  noticed,  is  not  usually  a  wave  set — the  only  ex- 
ception arising  when  the  level  surfaces  are  parallel.     For  a  given 


58  THE   PRINCETON   COLLOQUIUM. 

medium  the  number  of  wave  sets  is  oo°°,  since  there  is  one  for 
each  surface.  Each  of  these  sets  is  cut  by  the  level  surfaces  in 
the  equidistant  curves  of  the  wave  set;  that  is,  along  any  one  of 
these  curves  the  distance  between  consecutive  wave  surfaces 
remains  the  same.* 

46.  A  single  set  of  wave  fronts  has  no  geometric  peculiarity. 
That  is,  given  any  set  of  surfaces  f(x,  y,  z)  =  constant,  it  will 
always  be  possible  to  find  a  medium  in  which  that  set  will  serve 
as  a  wave  set.  In  fact  there  are  oo00  such  media.  For  in 
equation  (4),  the  given  function  /,  without  altering  the  given 
surfaces,  may  be  replaced  by  an  arbitrary  function  &(/)  of  itself, 
and  this  gives  oo°°  distinct  values  for  v. 

When  will  two  sets  of  wave  fronts  be  consistent?  Two  arbi- 
trary sets  of  surfaces  /  =  constant,  /i  =  constant  cannot  usually 
be  regarded  as  wave  sets  in  any  single  medium.  The  requisite 
condition  is 


fil  +  fif+fif      Oi(/i)f 

where  ft,  fti  may  be  any  functions.  An  equivalent  condition 
is  that  it  must  be  possible  to  chose  parameters  for  the  two  sets 
in  such  a  way  that 

df_  =  dj± 

dn      dn\ 

where  dn  and  dn\  denote  the  normal  distance  between  consecutive 
surfaces. 

47.  But  a  clearer  answer  may  be  given  in  terms  of  the  geometric 
properties  A  and  B.  If  a  set  of  surfaces  is  to  be  a  wave  set,  the 
oo2  orthogonal  curves  must  be  members  of  the  natural  family  of 
oo4  rays.  If  two  sets  of  wave  fronts  are  given,  we  have  then  two 
congruences  of  curves.  The  question  then  is,  when  can  two 
normal  congruences  of  curves  be  regarded  as  belonging  to  a 
natural  family? 

*  This  follows  immediately  from  (4).  It  is  to  be  remarked,  however,  that 
this  property  is  not  characteristic  of  wave  sets. 


ASPECTS   OF   DYNAMICS.  59 

Take  any  point  p  in  space,  and  consider  the  two  curves,  one 
from  each  of  the  congruences,  passing  through  it.  The  circles 
of  curvature  at  p  must  intersect  again  at  some  point  P  (by 
property  A).  This  condition  makes  sure  that  the  two  congru- 
ences belong  to  some  velocity  system.  If  now  this  is  to  be  a 
natural  system,  we  must  also  add  property  B  or  rather,  since 
no  hyperosculating  circles  are  directly  defined,  the  equivalent 
restriction  (see  page  47)  relating  to  the  transformation  from  p  to 
P.  The  final  answer  may  then  be  given  as  follows: 

Two  sets  of  icave  surfaces  belong  to  the  same  optical  medium  when 
and  only  when  they  satisfy  the  following  geometric  conditions : 

(A')  At  any  point  p  of  space  the  circles  of  curvature  of  the  orthog- 
onal trajectories  of  the  two  sets  of  surfaces,  passing  through  that 
point,  intersect  again  at  some  point  P. 

(B'}  The  point  transformation  from  p  to  P  has  the  property  that 
the  three  lineal  elements  of  p  each  of  which  corresponds  to  a  cocircular 
element  at  P  are  mutually  orthogonal. 

48.  Two  sets  of  surfaces  taken  at  random  will  not  belong,  as 
wave  sets,  to  any  medium.     On  the  other  hand,  as  we  have  said, 
one  set  belongs  to  <x>  °°  distinct  media.     The  question  then  arises, 
just  what  will  uniquely  determine  a  medium. 

A  natural  family  is  uniquely  determined  if  we  are  given  one  set 
of  wave  fronts  and  a  single  extra  trajectory.  This  means  a  tra- 
jectory not  belonging  to  the  congruence  defined  as  the  orthogonal 
trajectories  of  the  wave  set. 

49.  The  extra  curve  however  cannot  be  taken  at  random;  it 
must  be  related  in  a  certain  way  to  the  wave  set.     If  the  wave 
set  is  f(x,  y,  z)  =  constant,  then  the  condition  on  the  curve  is 
that  it  satisfy  the  Monge  equation  of  second  order 


(3) 


2Az"  —  fl  4-  v'2  4-  z'VA    —  z'A  ")  /  —  z'f 

»""**  V  X         I         €*  J        Ai        /  i  _A  j  /w    _i   -  ./  Z  &  J  X 

where 

(3')  As /,*+/,*+/,*. 


60  THE  PRINCETON  COLLOQUIUM. 

Here/,  and  hence  A,  are  given,  and  y  and  z  are  unknown  functions 
of  x.  The  interpretation  is  obvious  from  property  A. 

In  order  that  an  extra  curve  shall  be  consistent  with  a  given 
wave  set  (that  is,  in  order  that  both  shall  belong  to  a  single 
medium)  it  is  necessary  and  sufficient  that  the  curve  shall  cross 
the  surfaces  (of  course  obliquely)  in  such  a  way  that  at  any  point 
of  intersection  the  circle  of  curvature  of  the  extra  curve  shall 
intersect  the  circle  of  curvature  of  the  curve  orthogonal  to  the 
surfaces.  When  the  curve  satisfies  this  restriction,  it  defines 
with  the  given  wave  set  a  unique  natural  family. 

50.  If  we  are  merely  given  one  wave  set,  the  number  of  possible 
media  is  oo"  (since  v  involves  arbitrary  functions).      Each  of 
these  has  oo4  rays  (forming  a  natural  family).     The  totality  of 
media  give  rise  to  a  totality  of  oo00  rays,  namely  the  solutions 
of  the  Monge  equation  of  second  order  (3).     This  equation  is  of 

the  type 

Ay"  +  Bz"  +  C  =  0 

(where  the  coefficients  are  functions  of  x,  y,  z,  y',  z'},  which  the 
author  has  shown  to  be  characterized  by  the  Meusnier  property:* 
Those  curves  which  pass  through  a  given  point  in  a  given  direction 
have  circles  of  curvature  (constructed  at  the  common  point) 
generating  a  sphere. 

51.  The   inverse   problem   connected   with   natural   families, 
namely,  given  the  oo4  trajectories  to  construct  the  generating  field 
of  force,  is  solved  immediately  in  connection  with  property  A. 
The  force  acting  at  any  point  p  acts  in  the  line  joining  that  point 
to  the  corresponding  point  P,  and  its  intensity  is  proportional  to 
the  reciprocal  of  the  distance  between  the  two  points,  f     This 
construction  may  be  carried  out  if  we  know  a  sufficient  number  of 
trajectories,  without  knowing  the  whole  system. 

52.  The  greatest  number  of  rays  which  two  distinct  media 

*Kasner,  Butt.  Amer.  Math.  Soc.,  vol.  14  (1908),  pp.  461-465.  The 
result  includes  the  extension  of  Meusnier's  theorem  made  by  Lie,  and  is  in 
fact  the  largest  generalization  possible. 

t  The  determination  of  the  potential  function  W(x,  y,  2)  or,  what  is  equiv- 
alent, the  index  of  refraction  v(x,  y,  2),  requires  a  quadrature. 


ASPECTS   OF   DYNAMICS.  61 

can  have  in  common  is  oo2  (one  through  each  point  of  space). 
If  two  media  have  that  many  in  common,  it  is  easily  shown  that 
the  resulting  congruence  is  necessarily  normal.  Any  normal 
congruence  can  be  obtained  in  this  way,  for,  as  stated  above,  it 
belongs,  not  only  to  two,  but  to  oo*°  distinct  media. 

53.  We  mention  only  one  special  problem:  the  determination  of 
those  media  in  which  disturbances  are  propagated  by  Lame  families 
of  surfaces ;  that  is,  every  wave  set  is  to  be  of  the  Lame  type  (thus 
forming  part  of  a  triply  orthogonal  family  of  surfaces).  The 
index  of  refraction  is  found  to  vary  inversely  as  the  power  of  the 
point  with  respect  to  a  fixed  sphere;  the  rays  then  are  the  oo4 
circles  orthogonal  to  that  sphere.  Since  the  radius  of  the  sphere 
may  be  zero,  real,  or  imaginary,  these  media  yield  well  known 
interpretations  of  parabolic,  hyperbolic,  and  elliptic  geometries. 
(See  Transactions  of  the  American  Mathematical  Society,  volume 
12  (1911),  pages  70-74.) 

§§  54-61.    A  SECOND  CONVERSE  PROBLEM  CONNECTED  WITH 
THE  THOMSON-TAIT  THEOREM 

54.  Consider  the  general  conservative  field,  defined  by  its 
work  function  W(x,  y,  z).  With  any  motion  of  the  particle  there 
is  associated  a  definite  value  of  the  constant  of  total  energy 

1V2  _  w  =  h. 

If  h  is  not  assigned  the  complete  system  of  trajectories  is  made  up 
of  oo5  curves. 

Consider  now  an  arbitrary  surface,  which  we  term  the  base 
surface, 

(2)  z  =  /Or,  y). 

From  each  of  its  points  we  may  draw  normal  to  the  surface  oo1 
trajectories  since  the  initial  value  of  the  speed  v  is  arbitrary. 
We  thus  have  in  all  oo3  trajectories  normal  to  2.  In  order  to  have 
a  congruence  we  must  assign  the  value  of  v  at  each  point  of  2, 
that  is,  we  must  give  a  law  of  distribution  of  the  initial  speed.  The 
question  arises:  What  form  of  law  will  make  the  corresponding 


62  THE   PRINCETON   COLLOQUIUM. 

congruence  a  normal  congruence?  Of  course  for  any  law  the 
congruence  will  be  orthogonal  to  the  base  surface,  but  usually 
it  admits  no  other  orthogonal  surfaces. 

The  Thomson-Tait  theorem  (in  its  complete  dynamical  form) 
gives  one  such  law:  it  states  that  if  the  initial  speed  is  selected  so  as 
to  make  h  have  the  same  value  at  all  the  points  of  2,  the  congru- 
ence will  be  normal.  It  thus  gives  a  plan  for  constructing  oo1 
normal  congruences  for  a  given  base,  one  for  each  value  of  h. 
We  shall  refer  to  any  one  of  these  as  "constructed  according 
to  the  Thomson-Tait  law." 

Is  this  the  only  answer  to  our  question?  If  oo2  trajectories 
are  drawn  orthogonal  to  2  and  if  they  form  a  normal  congruence, 
does  it  follow  that  the  distribution  of  values  of  the  initial  speed 
is  precisely  such  that  the  sum  of  the  kinetic  and  potential  energies 
has  the  same  value  at  all  points  of  2? 

The  requisite  discussion  is  not  simple.  We  shall  merely  state 
the  results  we  have  obtained. 

55.  The  answer  to  our  question  is  "  in  general  "  in  the  affirma- 
tive.    The  first   converse  theorem,  discussed  in    §  37,  is  true 
without  exception.     The  present  is  true  with  exceptions  —  which 
may  be  definitely  limited. 

For  a  "  general  "  base  surface  2  in  a  given  conservative  field  of 
force,  the  only  congruences,  formed  by  oo2  trajectories  orthogonal  to 
2  (one  draicn  at  each  point),  which  admit  an  infinitude  of  orthogonal 
surfaces,  are  those  constructed  according  to  the  Thomson-Tait  law 
(so  that  the  total  energy  has  a  constant  value). 

56.  To  make  this  precise  we  must  of  course  limit  the  class  of 
exceptional  surfaces  connected  with  a  given  field.     These  appear 
in  the  analytical  discussion  as  the  solutions  of  a  certain  partial 
differential  equation  of  the  second  order* 


_ 

Wx  +  qWz      Wx  +  qW. 


*  The  expanded  result  is  of  the  form 

PIT  +  Pzs  +  P3t  +  P4=Q, 
where  r,  s,  t  denote  the  derivatives  of  second  order  of  z  =  f(x,  y). 


ASPECTS  OF  DYNAMICS.  63 

where  W  is  the  given  work  function,  and 

w  =  pWx  +  qW,  -  W, 

Vi  +  PZ  +  ?2 

This  differential  equation  defines  a  class  of  surfaces  which  is 
seen  to  depend  only  on  the  equipotential  surfaces 

W(x,  y,  z)  =  constant. 

The  result  may  be  put  into  geometric  form  and  stated  as  follows : 
The  only  surfaces  2  which  may  be  exceptional  in  the  theorem  of  §  55 
(that  is,  which  may  give  rise  to  normal  congruences  not  included  in 
the  Thomson-Tait  law}  are  those  with  this  property:  along  each  of 
the  equipotential  lines*  of  the  surface  the  component  of  the  acting 
force  normal  to  the  surface  is  constant. 

57.  Observe  that  it  is  not  stated  that  the  surfaces  described, 
which  exist  in  any  field,  actually  give  rise  to  additional  normal 
congruences.  To  understand  the  situation  more  precisely,  it  is 
necessary  to  observe  that  in  the  analytic  discussion  the  condition 
for  a  normal  congruence  is  developed  in  the  form 

i  Q!  +  *2ft2  +  •  •  •  =  0, 

where  t  is  the  parameter  which  varies  along  the  curve,  starting 
with  the  value  zero  on  the  surface  2,  and  the  coefficients  12  are 
functions  of  the  two  parameters  defining  the  initial  points  on  2. 
By  assumption  the  congruence  is  orthogonal  to  2,  so  the  term  Q0» 
independent  of  t,  will  not  appear.  For  a  normal  congruence  all 
the  coefficients  ft  must  vanish.  If  only  a  certain  number  vanish 
the  congruence  may  be  described  as  approximately  normal  (the 
approximation  being  of  degree  n  if  fti  =  ft2  •  •  •  =  ftn  =  0) : 
the  curves  are  then  orthogonal  not  only  to  2  but  also  to  one  or 
more  (infinitesimally)  adjacent  surfaces. 

58.  If  now  we  impose  on  the  congruence  of  trajectories  normal 
to  2  the  condition  fti  =  0,  we  find  that  this  may  be  fulfilled  for 

*  The  equipotential  lines  of  any  surface  are  the  lines  cut  out  by  the  equi- 
potential surfaces  W  =  const. 


64  THE   PRINCETON  COLLOQUIUM. 

any  surface:  the  restriction  is  merely  on  the  law  of  initial  speed 
and  means  that  the  total  energy  must  be  the  same,  not  necessarily 
over  the  entire  surface,  but  along  each  equipotential  line  of  the 
surface.* 

59.  If  we  further  impose  the  condition  Q2  =  0,  then  for  a 
"  general  surface  "  the  law  of  speed  must  be  the  Thomson-Tait 
law,  but  for  an  "  exceptional  surface  "  the  law  is  the  more  general 
one  just  stated. 

60.  The  discussion  of  the  higher  conditions  123  =  0,  etc.,  we 
have  not  completed.     It  is  therefore  not  known  precisely  in 
which  cases  normal  congruences  (in  the  exact  sense)  may  arise. 
For  central  and  parallel  fields  it  may  be  shown  that  the  exceptional 
surfacesf  actually  give  rise  to  normal  congruences  (in  addition 
to  those  included  in  the  Thomson-Tait  theory):  for  such  fields 
the  vanishing  of  the  higher  coefficients  follows  from  the  vanishing 
of  the  first  two. 

61.  The  principal  results  of  the  converse  problem  may  be 
formulated  as  follows: 

//  oo 2  trajectories  (of  a  conservative  field),  meeting  a  surface  2 
orthogonally,  are  also  orthogonal  to  an  infinitesimally  adjacent 
surface,  then  the  total  energy  along  each  equipotential  line  of  S 
is  constant. 

If  oo 2  trajectories,  selected  from  the  complete  system  of  oo 5,  form  a 
normal  congruence,  then  in  general  they  will  all  belong  to  the  same 
natural  family  (that  is,  the  total  energy  will  be  the  same  for  all  the 
curves);  except  possibly  when  the  oo1  orthogonal  surfaces*  are  ex- 
ceptional in  the  sense  defined  in  §  56  (the  additional  congruences 
then  and  only  then  are  normal  to  at  least  the  second  degree  of 
approximation). 

Normal  congruences  not  of  the  Thomson-Tait  type  (that  is,  not 


*  If,  in  particular,  the  surface  is  one  of  the  equipotential  surfaces,  the  dis- 
tribution of  speed  is  thus  entirely  arbitrary. 

t  In  the  case  of  ordinary  constant  gravity  the  exceptional  surfaces  are 
those  termed  moulure  surfaces  by  Monge:  they  are  generated  by  rolling  the 
plane  of  any  plane  curve  about  a  vertical  cylinder  of  arbitrary  cross  section. 

J  If  one  of  these  surfaces  is  exceptional,  all  will  be. 


ASPECTS   OF    DYNAMICS.  65 

selected  from  within  a  natural  family)  actually  arise  for  central 
and  parallel  fields. 

§§  62-67.  GEOMETRIC  FORMULATION  OF  SOME  CURIOUS  OPTICAL 

PROPERTIES 

62.  In  Thomson  and  Tait's  Natural  Philosophy*  the  character- 
istic function  of  Hamilton  is  applied  to  the  motion  of  a  particle 
in  a  conservative  field  of  force,  and  certain  results  are  obtained 
which  we  shall  try  to  restate  as  purely  geometric  properties  of 
a  natural  family  of  trajectories.  To  what  extent  these  properties 
are  characteristic  is  not  settled.  We  quote  the  principal 
passages  referred  to. 

"  Let  two  stations,  0  and  0',  be  chosen.  Let  a  shot  be  fired 
with  a  stated  velocity,  V,  from  0,  in  such  a  direction  as  to 
pass  through  0'.  There  may  clearly  be  more  than  one  nat- 
ural path  by  which  this  may  be  done;  but,  generally  speaking, 
when  one  such  path  is  chosen,  no  other,  not  considerably  diverging 
from  it,  can  be  found;  and  any  infinitely  small  deviation  in  the 
line  of  fire  from  0,  will  cause  the  bullet  to  pass  infinitely  near  to, 
but  not  through,  0'.  Now  let  a  circle,  with  infinitely  small 
radius  r,  be  described  round  0  as  center,  in  a  plane  perpendicular 
to  the  line  of  fire  from  this  point,  and  let — all  with  infinitely  nearly 
the  same  velocity,  but  fulfilling  the  condition  that  the  sum  of  the 
potential  and  kinetic  energies  is  the  same  as  that  of  the  shot  from  0 
—bullets  be  fired  from  all  points  of  this  circle,  all  directed  infinitely 
nearly  parallel  to  the  line  of  fire  from  0,  but  each  precisely  so  as 
to  pass  through  0'.  Let  a  target  be  held  at  an  infinitely  small 
distance,  a',  beyond  0',  in  a  plane  perpendicular  to  the  line  of  the 
shot  reaching  it  from  0.  The  bullets  fired  from  the  circum- 
ference of  the  circle  round  0,  will,  after  passing  through  0' ', 
strike  this  target  in  the  circumference  of  an  exceedingly  small 
ellipse,  each  with  a  velocity  (corresponding  of  course  to  its 
position,  under  the  law  of  energy)  differing  infinitely  little 
from  V,  the  common  velocity  with  which  they  pass  through  0'. 
Let  now  a  circle,  equal  to  the  former,  be  described  round  0', 

*Part  I  (Cambridge,  1903),  pp.  355-359. 
13 


66  THE   PRINCETON   COLLOQUIUM. 

in  the  plane  perpendicular  to  the  central  path  through  0',  and 
let  bullets  be  fired  from  points  in  its  circumference,  each  with 
the  proper  velocity,  and  in  such  a  direction  infinitely  nearly 
parallel  to  the  central  path  as  to  make  it  pass  through  0.  These 
bullets,  if  a  target  is  held  to  receive  them  perpendicularly  at  a 
distance  a  =  a'V/V,  beyond  0,  will  strike  it  along  the  circum- 
ference of  an  ellipse  equal  to  the  former  and  placed  in  a  "  cor- 
responding "  position;  and  the  points  struck  by  the  individual 
bullets  will  correspond;  according  to  the  following  law  of  "  cor- 
respondence ": — Let  P  and  P'  be  points  of  the  first  and  second 
circles,  and  Q  and  Q'  the  points  of  the  first  and  second  targets 
which  bullets  from  them  strike;  then  if  P'  be  in  a  plane  containing 
the  central  path  through  0'  and  the  position  which  Q  would 
take  if  its  ellipse  were  made  circular  by  a  pure  strain;  Q  and  Q' 
are  similarly  situated  on  the  two  ellipses." 

63.  The  second  passage  is  as  follows :  "  The  most  obvious  optical 
application  of  this  remarkable  result  is,  that  in  the  use  of  any 
optical  apparatus  whatever,  if  the  eye  and  the  object  be  inter- 
changed without  altering  the  position  of  the  instrument,  the  mag- 
nifying power  is  unaltered."  ..."  Let  the  points  0  and  0'  be  the 
optic  centers  of  the  eyes  of  two  persons  looking  at  one  another 
through  any  set  of  lenses,  prisms,  or  transparent  media  arranged 
in  any  way  between  them.  If  their  pupils  are  of  equal  size  in 
reality,  they  will  be  seen  as  similar  ellipses  of  equal  apparent 
dimensions  by  the  two  observers.  Here  the  imagined  particles 
of  light,  projected  from  the  circumference  of  the  pupil  of  either 
eye,  are  substituted  for  the  projectiles  from  the  circumference 
of  either  circle,  and  the  retina  of  the  other  eye  takes  the  place 
of  the  target  receiving  them,  in  the  general  kinetic  statement."* 

*  This  fact  and  many  other  applications  are  included  in  the  following 
general  proposition.  "  The  rate  of  increase  of  any  one  component  momentum, 
corresponding  to  any  one  of  the  coordinates,  per  unit  of  increase  of  any  other 
coordinate,  is  equal  to  the  rate  of  increase  of  the  component  momentum  cor- 
responding to  the  latter  per  unit  increase  or  dimension  of  the  former  coordinate, 
according  as  the  two  coordinates  chosen  belong  to  one  configuration  of  the 
system,  or  one  of  them  belongs  to  the  initial  configuration  and  the  other  to 
the  final." 


ASPECTS  OF  DYNAMICS.  67 

64.  The  statement  in  the  first  passage  is  not  purely  geometric; 
for  it  involves  not  only  the  curves  described,  but  also  the  speeds 
V  and  V  at  the  points  0  and  0'.  We  therefore  try  to  formulate 
the  part  of  the  theorem  which  is  really  geometric. 

We  have  a  natural  family  made  up  of  <»4  curves  in  space, 
one  for  each  initial  lineal  element  (point  and  direction)  of  space. 
Select  any  one  of  these  curves  c  and  any  two  points  0  and  0' 
upon  it.  Construct  the  planes  p  and  p'  normal  to  this  curve  at 
0  and  0'. 

For  each  direction  through  0,  a  curve  of  our  family  is  deter- 
mined; this  strikes  the  plane  p'  at  a  definite  point.  We  thus  have 
a  certain  correspondence  between  the  bundle  of  directions 
through  0  and  the  points  of  p'.  For  directions  infinitesimally 
close  to  the  direction  of  c  at  0,  and  for  points  close  to  0',  this 
correspondence  is  linear;  and  by  a  proper  selection  of  cartesian 
axes  at  0  and  0',  we  may  write  the  correspondence  in  the  canon- 
ical form 


where  (xr,  y'}  denote  the  coordinates  of  the  point  in  the  plane 
p',  and  the  corresponding  direction  at  0  has  direction  cosines 
proportional  to  (£  :  77  :  1). 

In  an  entirely  analogous  way,  by  considering  the  curves  of 
the  natural  family  which  go  through  0',  and  the  points  of  inter- 
section with  the  plane  p,  we  obtain  a  second  linear  correspond- 
ence which  may  be  reduced  to  the  form 

£'  =  <xzx,     77'  =  /%, 

where  (x,  y)  is  the  point  in  the  plane  p  and  (£'  :  77'  :  1)  gives  the 
corresponding  direction  at  0'. 

If  we  were  dealing  with  an  arbitrary  family  of  °o  4  curves,  in- 
stead of  a  natural  family,  these  linear  correspondences  would  still 
exist;  but  the  choice  of  axes  in  the  second  canonical  form  would 
be  different  from  that  required  in  the  first,  and  the  two  constants 
appearing  in  the  second  form  would  be  independent  of  those 


68  THE   PRINCETON  COLLOQUIUM. 

appearing  in  the  first.  The  peculiarity  of  the  natural  type  may  be 
stated  in  the  folloicing  form:  First,  the  canonical  axes  for  the  two 
correspondences  coincide;  second,  the  ratio  of  the  characteristic 
constants  has  the  same  value  for  both  correspondences. 

This  is  the  essential  geometric  content  of  the  long  statement 
quoted  above  from  Thomson  and  Tait.  Is  this  characteristic 
of  the  natural  type?  We  do  not  know. 

64'.  A  statement  in  more  concrete  terms  is  of  interest.  If  we 
start  out  from  0  in  directions  equally  inclined  (the  fixed  angle 
is  of  course  assumed  infinitesimal)  to  the  direction  of  c, 
that  is,  along  a  cone  of  revolution  having  for  axis  the  tangent 
of  c,  the  resulting  trajectories  forming  a  sort  of  curvilinear  cone, 
we  strike  points  on  p'  located  on  an  ellipse  with  0'  as  center. 
By  changing  the  angle  of  the  cone  we  obtain  a  family  of  similar 
and  similarly  situated  ellipses.  The  principal  axes  of  these 
ellipses  are  the  canonical  directions  referred  to  above  for  the  first 
correspondence,  and  the  ratio  of  the  diameters  is  equal  to  the 
ratio  of  the  canonical  constants  (a\  :fi\).  By  starting  from  the 
other  point  0'  along  cones  of  revolution  having  for  axis  the  tan- 
gent to  c,  we  strike  the  plane  p  in  a  second  set  of  homothetic 
ellipses.  The  two  sets  of  ellipses  thus  obtained,  one  in  the  plane  p, 
and  the  other  in  the  plane  p',  are  similar.  This  is  part  of  the 
property  stated,  but  not  the  whole.  It  should  be  observed  that 
it  has  no  meaning  to  say  that  the  two  sets  are  similarly  situated, 
since  they  are  in  different  planes. 

65.  We  may,  however,  obtain  two  sets  in  the  same  plane  as 
follows:  If  we  start  along  the  cone  of  revolution  from  0,  we  hit  p' 
in  an  ellipse.  If  we  wish  to  hit  p  in  a  circle,  we  must  start  at  0' 
along  a  certain  elliptical  cone:  the  sections  of  this  cone  by  planes 
parallel  to  p',  projected  orthogonally  on  p',  give  a  set  of  homothetic 
ellipses.  We  thus  have  in  the  plane  p',  two  sets  of  ellipses,  the 
first  set  being  obtained  from  cones  of  revolution  at  0,  and  the 
second  set  being  obtained  from  elliptical  cones  at  0'  by  orthogonal 
projection  of  parallel  sections.  If  we  were  dealing  with  an  arbi- 
trary family  of  curves,  the  two  sets  thus  obtained  would  be  un- 
related :  for  a  natural  family,  however,  the  two  sets  coincide. 


ASPECTS   OF  DYNAMICS.  69 

66.  Of  course  we  could  also  construct  two  sets  in  the  plane  p 
and  these  would  coincide;  but  this  would  not  give  an  additional 
property.     In  the  statement  quoted,  certain  pairs  of  congruent 
instead  of  merely  similar  ellipses  appear,  but  that  is  due  to  the 
introduction  of  kinematics:  namely,  use  is  made  of  the  velocities 
V  and  V  at  the  points  0  and  0'.     "  If  0  and  0'  are  regarded  as 
optic  centers  of  the  eyes  of  two  persons  looking  at  one  another 
through  any  optical  apparatus,  and  if  their  pupils  are  of  equal 
size  in  reality,  they  will   be  seen  as  similar  ellipses  of  equal 
apparent  dimensions  by  the  two  observers."     It  should  be  ob- 
served, however,  that  the  dimensions  will  be  equal  only  under 
the  assumption  that  the  two  eyes  are  at  positions  for  which  the 
velocities  V  and  V,  or,  what  is  equivalent,  the  indices  of  re- 
fraction v  and  v',  are  equal.     In  the  most  general  case  of  an 
isotropic  medium,  the  ellipses  will  not  have  equal  apparent  di- 
mensions, but  the  ratio  of  the  dimensions  will  be  equal  to  the 
ratio  of  the  two  velocities. 

67.  TWTO  converse  questions  remain  unanswered.     First:  Find 
all  systems  of  oo4  curves  in  space  such  that  circles  about  0  and 
0'  appear  as  similar  ellipses. 

Second:  Find  all  systems  such  that  the  set  of  ellipses  in  the 
plane  p'  formed  by  starting  from  0  along  cones  of  revolution, 
and  the  set  of  ellipses  found  by  orthogonal  projection  upon  p' 
of  the  sections  cut  out  by  planes  parallel  to  p'  of  those  (elliptical 
curvilinear)  cones  at  0'  which  strike  plane  p  in  circles, — such  that 
these  two  sets  of  ellipses  shall  coincide. 

§§  68-72.     THE  SO-CALLED  GENERAL  PROBLEM  OF  DYNAMICS 

68.  Consider  any  material  system  (particles  or  rigid  bodies) 
with  n  degrees  of  freedom,  so  that  its  position  at  each  instant  is 
determined  by  n  independent  coordinates  denoted  by  x\,  x2,  -  •  • , 
xn.     The  kinetic  energy  T  will  be  represented  by  a  quadratic  form 

2T  =  ^ctikXiXk, 
where   the  coefficients  a  are  functions  of   the  coordinates,  and 


70  THE   PRINCETON   COLLOQUIUM. 

the  dots  denote  time  derivatives.  If  the  acting  forces  are  con- 
servative, there  will  exist  a  force  function  W(xit  x2,  •••,£«), 
which  is  assumed  to  be  independent  of  the  time,  and  the  equation 
of  energy 

T  -  W  =  h 

asserts  that  in  any  given  motion  the  sum  of  the  kinetic  and 
potential  energies  is  constant. 

The  so-called  general  problem  of  dynamics  requires  the 
determination  of  the  motions  when  we  are  given  the  form  T, 
the  function  W,  and  the  constant  h.  The  possible  trajectories 
are  then  given  by  the  Jacobi  principle  of  least  action  as  the 
extremals  of  the  integral 


This  defines  the  most  general  natural  family.  The  integral  is  of  the 
form  jFds,  where  F  is  any  point  function  and  ds  is  the  length- 
element  in  a  general  n-dimensional  variety  Vn  defined  by 

ds2  =  Zaikdxidxk. 

69.  Such  a  family  consists  of  ccz(-n~u  curves,  in  the  space  Vn, 
one  passing  through  each  point  in  each  direction.  A  complete 
characterization  is  given  by  J.  Lipke,  in  his  doctor's  dissertation,* 
as  follows : 

(Ai)  The  locus  of  the  centers  of  geodesic  curvature  of  the  oo  n~"1 
curves  passing  through  any  point  of  Vn  is  a  flat  space  of  n  —  1 
dimensions  Sn-\- 

(A2)  The  osculating  geodesic  surfaces  (two-dimensional 
varieties)  at  the  given  point  form  a  bundle  of  surfaces,  all  con- 
taining a  fixed  direction  (and  hence  the  geodesic  line  in  that 
direction)  which  is  normal  to  the  Sn-i  of  property  AI. 

(B)  The  n  directions  at  any  point,  in  which,  as  a  consequence 
of  the  preceding  properties,  the  osculating  geodesic  circles  (circles 

*  Trans.  Amer.  Math.  Soc.,  vol.  13  (1912),  pp.  77-95. 


ASPECTS   OF   DYNAMICS.  71 

of  constant  geodesic  curvature)  hyperosculate  the  curves  of  the 
given  family,  are  mutually  orthogonal. 

70.  This  gives  the  generalization  of  properties  A  and  B  stated 
in  §§  29-31.     The  simpler  results  there  given  for  ordinary  space 
apply  to  a  euclidean  space  of   any  dimensionality  and  also  to 
spaces  of  constant  curvature.     In  the  general  space  of  variable 
curvature,  the  geodesic  circles  constructed  at  a  given  point  do 
not  all  meet  at  a  second  point,  and  so  no  analogue  of  the  law  of 
reciprocity  of  natural  families  presents  itself. 

71.  The  theorem  of  Thomson  and  Tait  remains  valid  for  any 
space.*     The  converse  questions  connected  with  it  have  not  been 
settled.     In  all  probability  the  Thomson-Tait  geometric  property 
is  characteristic  in  any  space  (flat  or  curved)  of  dimensionality 
greater  than  two.     Obviously  in  the  case  of  two  dimensions  the 
geometric  converse  is  not  valid,  since  any  system  of  oo1  curves 
admits  oo1  orthogonal  curves. 

72.  The  systems  characterized  by  property  A  (meaning  A\ 
together  with  AZ)  are  the  most  general  velocity  systems  in  Vn. 
The  case  n  =  2  presents  a  peculiar  feature:  for  then,  included  in 
the  velocity  type,  we  have,  in  addition  to  the  natural  type,  another 
special  type  of  interest  (geometric,  rather  than  dynamic),  namely 
the  isogonal  typef  (systems  formed  by  the  oo 2  isogonal  trajectories 
of  an  arbitrary  simply  infinite  system  of  curves).     In  the  case 
of  the  plane  (or  any  surface  of  constant  curvature)  the  reciprocity 
construction  for  velocity  systems  is  available,  and  each  of  the 
species,    natural    and    isogonal,    is    self-reciprocal.     The    only 
families  common  to  the  two  species  are  those  formed  by  the 
isogonals  of  an  isothermal  system,  or,  what  is  the  same,  by  velocity 
systems  generated  by  Laplacian  fields  of  force. * 

*  Cf.  Darboux,  Lemons,  vol.  2,  last  chapter,  where  references  to  the  memoirs 
of  Lipschitz  and  Beltrami  are  given. 

t  Scheffers  introduced  the  systems  of  plane  curves  y"  =  (\j/  —  y'<p) 
(1  +  y'2)  in  connection  with  the  theory  of  isogonals,  and  obtained  a  law  of 
reciprocity  for  isogonal  systems.  Cf.  Leipziger  Berichte,  1898,  1900;  Mathe- 
matische  Annalen,  vol.  60. 

J  Cf.  the  author's  note,  "  Isothermal  systems  in  dynamics,"  Butt.  Amer. 
Math.  Soc.,  vol.  14  (1908),  pp.  169-172. 


72  THE   PRINCETON   COLLOQUIUM. 

"We  note  finally  this  characteristic  distinction  between  the  two 
noteworthy  species: 

For  both  natural  and  isogonal  families  in  the  plane,  the  circles 
of  curvature  constructed  at  any  point  p  have  another  point  P 
in  common.  The  point  transformation  T  (from  p  to  P)  in  the 
natural  case  is  such  that  the  two  lineal  elements  at  any  point, 
each  of  which  is  converted  into  a  cocircular  element,  are  orthog- 
onal; while  in  the  isogonal  case  the  two  elements,  each  of  which 
is  converted  into  an  element  normal  to  a  cocircular  element,  are 
orthogonal. 

If  the  transformation  T  connected  with  a  velocity  system  is 
required  to  be  (direct)  conformal,  the  corresponding  field  must 
be  Laplacian.  Such  fields  are  distinguished  from  all  others  by 
the  fact  that  each  of  the  infinitude  of  systems  of  velocity  curves 
is  then  expressible  linearly  in  the  two  parameters  involved. 


CHAPTER  III 

TRANSFORMATION   THEORIES   IN  DYNAMICS 

§§  73-81.     PROJECTIVE  TRANSFORMATIONS 

73.  The  general  object  of  a  transformation  theory  is  to  relate 
new  problems  to  old  problems,  and  so  to  proceed  from  the  solution 
of  the  latter  to  the  solution  of  the  former.     The  most  important 
geometric  transformations  are  the  projective  and  the  conformal. 
Both  groups  play  important  roles  in  dynamics,  the  former  in 
connection  with  general  fields,  and  the  latter  in  connection  with 
conservative  fields. 

74.  The  importance  of  projective  transformations  in  dynamics 
was  brought  out   by  Appell    in    1889.     Given    any  positional 
field  of  force  in  the  plane,  the  corresponding  equations  of  motion 
are  of  the  form 

d?x  d?y 

(i)  ^2  =  ?fo  y)>      ^2  =  iK*i  y)' 

If  an  arbitrary  point  transformation,  unaccompanied  by  any 
change  in  the  time,  is  applied,  the  new  differential  equations 
will  usually  involve  not  only  x  and  y,  but  also  the  velocity  com- 
ponents dxfdt,  dyfdt.  In  fact  the  only  exception  is  where  the 
point  transformation  is  merely  affine: 

.1-1  =  ax  +  by  +  c,     yi  =  a'x  +  Vy  +  c'. 
Appell  showed  that  if  a  general  collineation 

(f)}  ax+  by  +  c  a'x  +  b'y  +  c' 

Xl  ~  a"x  +  b"y  +  c'"        yi  ~  a"x  +  V'y  +  c" 

is  accompanied  by  a  change  of  the  time  of  the  form 

at 


(2')  dt,  = 


k(a"x+  V'y  +  c")2' 
73 


74  THE   PRINCETON   COLLOQUIUM. 

the  new  differential  equations  will  be  of  the  original  form 

d?Xi  ? 

(3)  -2 


and  therefore  define  motion  in  some  new  positional  field  of  force. 
The  relation  between  the  new  field  and  the  original  field  is 
explicitly  as  follows 


*i  =  V(a"x  +  b"y  +  c"r-{C'W 
1    ft  =  P(a".r  +  b"y  +  c")2{  -  C(x}  -  y<p}  -  B<p 


where  the  capital  letters  denote  minors  in  the  determinant 
\ab'c"\  of  (2). 

74'.  The  trajectories  of  the  original  field  are  converted  by  the 
collineation  into  the  trajectories  of  the  new  field.  Also  the 
directions  of  forces  of  the  two  fields  are  protectively  related. 
It  must  not  be  thought,  however,  that  the  force  vector  acting 
at  a  given  point  (x,  y)  in  the  first  plane  is  projected  into  the 
new  force  vector  acting  at  the  point  (xi,  y\]  of  the  second  plane: 
the  initial  points  of  the  two  vectors  will  correspond,  of  course, 
by  the  given  collineation,  but  the  terminal  points  will  not.  The 
question  therefore  arises,  what  is  the  geometric  relation  between 
the  new  vector  field  and  the  old  vector  field? 

To  answer  this  question  we  take  our  rectangular  axes  so  that 
the  collineation  takes  its  metrical  normal  form.  (Affinities  of 
course  require  a  separate  discussion.)  The  canonical  formulas 
for  our  transformation  are 


x\  = 
(5) 

<Pi  =  —  tfyyia?<p,    ti  =  kzy1xi(x^  —  y<p), 
together  with 

(5')  dtl  =  ^' 

To  each  collineation  between  the  two  planes  corresponds  a  defi- 


ASPECTS   OF   DYNAMICS.  75 

nite  vector  transformation.  The  vectors  are  here  of  the  third 
type  (bound  vectors)  described  in  the  Introduction,  requiring 
four  coordinates  for  their  determination.  The  original  vector  is 
defined  by  the  four  numbers  (x,  y,  <f>,  ^),  the  first  two  defining 
the  initial  point,  and  the  last  two  giving  the  components  of  the 
vector.  The  coordinates  of  the  new  vector  are  (x\,  y\,  <p\,  *f/\). 
The  vector  transformation  induced  by  the  given  collineation  is 
not  projective.  The  new  vector  has  the  same  initial  point  and 
the  same  direction  as  the  projection  of  the  old  vector,  but  has  a 
different  length.  The  ratio  X  between  the  actual  length  of  the 
new  vector  and  the  length  of  the  projected  vector  is 

(5")  X  =  kW(x  +  <p). 

Noting  that  in  the  canonical  form  x  and  x-\-  <p  denote  the  distances 
from  the  initial  and  terminal  points  of  the  original  vector  to  the 
vanishing  line  in  the  first  plane,  we  may  state  this  result. 

Any  given  (non-affine*)  collineation  (2)  induces  a  certain  vector 
transformation  (determined  up  to  the  factor  k)  defined  analytically 
by  (2)  and  (4),  and  geometrically  as  follows:  If  PQ  is  any  bound 
vector  in  the  first  plane,  and  if  the  collineation  converts  the  initial 
point  P  into  PI  and  the  terminal  point  Q  into  Q\,  then  the  trans- 
formed bound  vector  is  not  PiQi,  but  PiQi,  where  Q\  is  the  point 
on  the  line  joining  PiQi  such  that  the  ratio  X  =  PiQi/PiQi  equals 
k2  times  the  cube  of  the  distance  from  P  to  the  vanishing  line  times 
the  distance  from  Q  to  that  vanishing  line. 

The  transformation  converts  the  <x> 4  bound  vectors  of  the  first 
plane,  represented  by  the  independent  coordinates  (x,  y,  <p,  \f/), 
into  the  oo 4  bound  vectors  of  the  new  plane,  f  In  the  dynamical 
application,  <p  and  \j/  are  given  as  functions  of  x,  y,  that  is,  we  have 
a  field  of  oo 2  vectors,  one  for  each  initial  point:  the  result  of 

*  In  the  case  of  an  affine  collineation,  the  induced  vector  transformation 
is,  except  for  the  constant  factor  k,  merely  the  result  of  applying  the  affinity 
to  both  ends  of  the  vector.  It  is  thus  linear. 

t  The  vector  transformations  induced  by  inverse  collineations  are  inverse 
to  each  other.  The  four-dimensional  transformations  are  therefore  Cremona 
transformations . 


76  THE   PRINCETON    COLLOQUIUM. 

the  transformation  is  a  new  field,  <p\  and  \f/i  being  expressible  in 
terms  of  Xi,  y\.  The  °o3  trajectories  of  the  first  field  are  con- 
verted by  the  collineations  into  the  <x>3  trajectories  of  the  new 
field;  it  is  to  be  noticed  however  that,  during  any  correspond- 
ing motions,  positions  which  correspond  according  to  the  col- 
lineation will  usually  not  correspond  to  the  same  instant  of 
time;  in  fact  from  (2') 

dt 


h 


J 


k2(a"x  +  b"y 


75.  If  X,  Y  denote  the  velocity  components  at  the  position 
x,  y  and  if  the  corresponding  velocity  in  the  second  plane  is 
Xi,  YI,  acting  at  the  position  x\,  y\,  then  we  find,  from  the  ca- 
nonical form  (5), 


,         Yl  =  ky^xY  -  yX). 

Thus  we  have  a  different  vector  transformation  which  may  be 
termed  the  phase*  transformation  (in  distinction  from  the  force 
transformation  of  §  74)  :  it  gives  the  relation  between  the  corre- 
sponding phases  in  the  two  planes. 

If  we  speak  of  points  and  vectors  which  correspond  in  the  two 
planes  according  to  the  given  collineation  as  projectively  related, 
then  the  result  may  be  stated  in  this  form: 

The  new  phase  vector  does  not  coincide  with  the  projection  of  the 
given  phase  vector:  it  has  the  same  initial  point,  but  the  ratio  of  the 
actual  length  to  the  length  of  the  projected  vector  is  k2  times  the  product 
of  the  distances  from  the  ends  of  the  original  vector  to  the  vanishing 
line  of  the  collineation. 

76.  Having  studied  the  Appell  transformation  and  its  geo- 
metric interpretation  in  terms  of  force  vectors  and  phase  vectors, 
we  now  ask  whether  other  more  general  transformations  can 
play  a  like  role.  Appell  proved  the  following  converse  theorem: 

*  The  phase  of  a  particle  at  any  instant,  in  the  sense  of  Gibbs,  is  its 
position  together  with  its  velocity:  it  is  defined  by  the  four  numbers  (x,  y,  x,  y). 


ASPECTS   OF   DYNAMICS.  77 

The  only  transformations  of  the  form 

xi  =  *(*,  y),        yi  =  V(x,  y),        dt^  =  n(x,  y)dt 
which  convert  every  set  of  differential  equations 
d2x  d?y 

(i)  ^  =  v(x,  y),      M  =  *(*,y), 

into  one  of  the  same  form  are  those  defined  by  (2),  (2'). 

77.  By  eliminating  the  time  from  (1),  giving  the  differential 
equation  of  the  trajectories  in  the  form  (page  7) 

(7)  (t  -  y'<p}y'"  =  {*,  +  (*„  -  vJy'  -  vyy'^y"  -  3<^"2, 

the  author  proved  that  the  only  point  transformations  which 
convert  every  trajectory  system  (of  a  positional  field)  into  a 
trajectory  system  are  the  collineations.  This  remains  valid 
even  in  the  domain  of  all  contact  transformations,  as  we  now 
proceed  to  show. 

We  first  consider  the  class  of  differential  equations  (cf.  page  11) 

(8)  y"'  =  G(x,  y,  y'}y"  +  H(x,  y,  y'}y'ft 

including  (7)  as  a  special  case,  and  characterized  geometrically  by 
the  possession  of  property  I  (that  is,  the  focal  locus  for  each  ele- 
ment is  a  circle  through  the  given  point) .  We  prove  this  theorem : 

The  only  contact  transformations  which  convert  every  equation 
of  type  (8)  (that  is,  every  system  of  curves  with  property  I)  into 
one  of  the  same  type  are  collineations  and  correlations. 

That  no  other  transformations  are  possible  is  seen  as  follows. 
If  a  contact  transformation  is  to  convert  type  (8)  into  itself,  it 
must  convert  the  part  common  to  all  systems  of  that  type  into 
itself.  The  curves  defined  by  y"  =  0,  that  is,  straight  lines, 
obviously  satisfy  (8)  for  every  form  of  G  and  H.  It  is  obvious 
that  no  other  (proper)  curves  satisfy  all  such  equations.  But 
since  we  are  dealing  with  contact  transformations  and  not  merely 
point  transformations,  we  must  replace  the  concept  curve  by 


78  THE  PRINCETON  COLLOQUIUM. 

the  concept  union.  In  the  plane  the  only  unions  which  are  not 
(proper)  curves  are  points.  A  point  is  regarded  as  made  up  of 
oo l  lineal  elements;  so  x  is  constant,  y  is  constant,  y'  is  ar- 
bitrary, and  therefore  y"  and  y'"  are  infinite.  Point  unions 
are  to  be  regarded  then  as  solutions  of  all  equations  (8).  The 
common  part  thus  consists  of  the  oo 2  straight  lines  and  the  oo 2 
points  of  the  plane.  If  this  is  to  go  into  itself,  either  points 
go  into  points  and  lines  into  lines,  or  else  points  go  into  lines  and 
lines  into  points.  We  thus  obtain  only  collineations  and  cor- 
relations. 

That  the  collineations  actually  leave  type  (8)  unchanged  is 
easily  verified  analytically.*  The  work  for  correlations  is  simpli- 
fied by  observing  that  every  correlation  may  be  reduced,  by 
means  of  collineations,  to  the  form  of  Legendre's  transformation 

(9)  xi  =  —  y',        yi  =  xy'  —  y,        yi  =  —  x, 

(which  is  simply  polarity  with  respect  to  the  conic  x2-\-2y— 1  =  0). 
Extending  (9),  we  find 

1  it'" 

0')  *"  -  fn      *'"  =  jp 

This  converts  equation  (8)  into  one  of  the  same  form 

(10)  7/x'"  =  Gi(xlt  yi,  yi'W  +  #iGn,  ylt  yi'}yi"\ 

the  new  coefficient  functions  being  related  to  the  old  as  follows: 

Gi  =  H(—  yi, 
Hi  =  G(—  yi, 

This  completes  the  proof  of  the  theorem  stated  on  the  previous 
page. 

78.  If  we  impose  property  II  on  the  system  (8),  that  is,  if  we 
consider  the  subclass  in  which 

(11)  H  =  ~r- 

y'  -  w(x,  y)' 


*  Trans.  Amer.  Math.  Soc.,  vol.  7  (1906),  p.  420. 


ASPECTS   OF  DYNAMICS.  79 

the  correlations  are  no  longer  available.  That  collineations 
actually  convert  this  subclass  into  itself  is  readily  verified. 
The  same  is  true  for  the  still  narrower  class,  characterized  by 
properties  I,  II,  and  III,  in  which  the  differential  equation  is  of 

the  form  (cf.  page  13) 

i 

(12)  (y'  -  «)</'"  =  |Xy'2  +  vtf  +  v\y"  +  Zy"\ 

79.  We  pass  now  to  the  case  of  dynamical  trajectories,  defined 
by  type  (7),  and  state  the  fundamental  result: 

Collineations  are  the  only  contact  transformations  of  the  plane 
which  convert  every  system  of  oo3  dynamical  trajectories  (belonging 
to  an  arbitrary  positional  field  of  force)  into  such  a  system. 

The  only  possibilities  here  also  are  collineations  and  corre- 
lations. The  former  actually  have  the  required  property. 
The  latter  have  not,  as  is  seen  by  observing  that  the  application 
of  the  Legendre  transformation  (9)  to  a  dynamical  equation  (8) 
will  result  in  a  new  equation,  which,  while  still  of  the  general 
form  (8),  will  not  usually  be  of  the  dynamical  form.* 

80.  Systems  of  trajectories  are  characterized  by  the  set  of  five 
geometric  properties  of  page  10.     Therefore  projective  transfor- 
mation will  convert  any  system  of  curves  having  these  properties 
into  a  system  having  the  same  properties.     So,  in  spite  of  the 
fact  that  the  properties  as  stated  involve  metric  ideas  (osculating 
parabolas,  angles,  circles  of  curvature,  etc.),  the  set  is  actually 
projectively  invariant.     It  ought  to  be  possible  therefore  to 
restate  the  geometric  characterization  in  projective  language. 

We  shall  not  attempt  to  carry  out  this  idea  completely,  and 
merely  restate  properties  I  and  II  as  follows: 

Consider  the  oo1  trajectories  passing  through  a  given  point  0 
in  a  given  direction  whose  slope  is  y'.  For  each  of  these  tra- 
jectories construct  the  conic  which  has  four-point  contact  at  0 
and  touches  the  line  determined  by  two  arbitrarily  selected 


*  We  see  from  (10')  that  the  coefficients  G  and  H,  which  are  rational  with  re- 
spect to  y',  are  converted  into  coefficients  which  are  not  usually  rational. 


80  THE   PRINCETON   COLLOQUIUM. 

points*  A  and  B  (which  remain  fixed  in  the  following  statements)  ; 
through  A  and  B  draw  tangents  to  the  conic  (in  addition  to  the 
fixed  line)  and  join  the  points  of  contact.  The  lines  thus  con- 
structed, one  for  each  of  the  <x>  l  trajectories,  will  form  a  pencil 
(property  I). 

As  the  initial  direction  (that  is  y'}  varies  about  0,  the  vertex  of 
the  pencil  just  described  will  move  along  a  straight  line]  passing 
through  0  (property  II). 

The  other  properties,  especially  the  fifth,  are  much  more 
complicated. 

81.  In  conclusion  we  point  out  another  way  in  which  the 
projective  group  enters  in  dynamics.  If  an  arbitrary  point 
transformation 

zi  =  3>(x,  y),        yi  =  V(x,  y) 

is  applied  to  the  differential  equations 

x  =  <p(x,  y),        y  =  ^(x,  y}, 

defining  motion  under  a  purely  positional  force,  the  newr  differ- 
ential equations,  of  the  more  general  form 

$*z  +  &vy  +  3>xxx2  +  23>xvxy  +  3>yyf 


will  usually  define  a  motion  due  to  a  positional  force  together 
with  a  force  depending  on  the  velocity  x,  y.  If  this  latter  force 
is  to  be  absent  the  transformation  will  be  affine,  as  already  re- 
marked (§  74).  If,  instead,  we  demand  that  the  latter  force 
shall  act  in  the  direction  of  the  velocity  (and  thus  be  in  the 
nature  of  a  resistance),  we  find  that  the  transformation  may 
be  any  collineation. 

More   generally,  projective  transformations  are  the  only  point 

*  In  the  original  metric  statements  these  are  of  course  the  circular  points 
at  infinity. 

t  The  force  direction  will  be  determined  protectively  as  the  harmonic  of 
this  line  with  respect  to  the  lines  joining  O  to  A  and  B- 


ASPECTS   OF   DYNAMICS.  81 

transformations  which  leave  invariant  the  type 

x  =  <p(x,  y}  -f  xR(x,  y,  x,  y), 
y  =  \f/(x,  y}  +  yR(x,  y,  x,  y), 

defining  motion  of  a  particle  under  any  positional  force  together  with 
any  resistance  term  acting  in  the  direction  of  motion. 

§§  82-91.       CONFORMAL   TRANSFORMATIONS 

82.  The  importance  of  conformal  transformation  is  well  known 
in  connection   with   the   theory   of  the  potential.     Geometric 
inversion  or  transformation  by  reciprocal  radii,  for  example, 
yields  the  method  of  electric  images  due  to  Sir  William  Thomson. 
In  connection  with  dynamics,  the  importance  of  general  con- 
formal  transformations  has  been  emphasized  by  Larmor,  Goursat, 
and  Darboux.* 

83.  Consider  any  conformal  representation  of  the  points  of 
two  surfaces  S  and  S\.     The  first  fundamental   forms  of  the 
surfaces  may  be  taken  to  be 

dsz  =  Edu2  +  2Fdudv  +  Gdv2, 
dsf  =  \(Edu*  +  2Fdudv  +  GW), 

where  corresponding  points  have  the  same  parameters  u,  v. 
The  principal  theorem  is  that  every  natural  system  on  one  surface 
becomes  by  the  conformal  representation  a  natural  system  on  the 
other.  This  is  obvious  if  we  remember  that  natural  systems  are 
obtained  by  minimizing  an  integral  in  which  the  integrand  is 
the  element  of  length  multiplied  by  any  point  function.  Hence 
The  only  point  transformations  (in  any  space}  which  convert 
every  natural  family  into  a  natural  family  are  the  conformal. 

84.  Consider  now  the  °o 3  dynamical  trajectories  on  S  produced 
by  a  conservative  field  of  force,  the  work  function  being  W '. 
These  consist  of  co1  natural  families,  one  for  each  value  of  the 

*  Cf.  the  discussion  in  Routh,   Dynamics   of   a    Particle,  Nos.  628-635 
•(method  of  inversion  and  conjugate  functions). 
14 


82  THE   PRINCETON   COLLOQUIUM. 

constant  of  total  energy  h.     It  will  be  convenient  to  refer  to  the 
particular  natural  system  produced  in  the  given  field  W  for  a 
particular  value  h,  as  the  family  due  to  W  +  h. 
The  corresponding  family  on  Si  is  due  to 

W+h 


Hence  the  oo  1  related  natural  families  on  S,  found  by  varying  h, 
go  over  by  the  conformal  representation  into  <x>  1  natural  families 
which  are  not  usually  related,  that  is,  do  not  form  the  complete 
system  of  trajectories  belonging  to  a  conservative  field.  The 
only  case  in  which  the  new  families  are  related  arises  when 

W  =  \, 

for  then  the  new  systems  are  due  to  the  work  function 

W,  =  1/X. 

We  then  reach  the  conclusion  that  in  any  conformal  representation 
(excluding  the  trivial  homothetic  case*)  there  is  a  unique  conservative 
force  whose  complete  system  of  oo3  dynamical  trajectories  is  con- 
verted into  the  complete  system  of  some  (usually  distinct)  conserva- 
tive force.  The  work  function  of  the  force  in  question  is  defined  by 
the  squared  ratio  of  magnification, 


85.  Similar  statements  may  be  made  for  brachistochrones. 
Every  system  of  oo2  brachistochrones  due  to  any  work  function 
and  a  given  value  of  h  of  course  becomes  such  a  system,  for  any 
natural  family  may  be  regarded  as  a  family  of  brachistochrones. 
But  there  is  only  one  complete  system  of  <x  3  brachistochrones  which 
is  converted  into  a  complete  system,  namely,  that  defined  by  the  work 

*  It  is  obvious  that  in  this  case  every  complete  system  of  trajectories  becomes 
a  complete  system.  The  same  holds  for  brachistochrones  and  catenaries. 


ASPECTS   OF  DYNAMICS.  83 

function 

W  =  1/X. 

For  any  other  work  function  the  oo  l  families  of  brachistochrones, 
due  to  W  +  h,  become  oo  :  non-related  natural  families  on  Si 
due  to 


86.  In  the  case  of  catenaries  due  to  W  +  h,  the  oo  1  usually 
non-related  natural  families  corresponding  on  Si  are  due  to 

W+h 

Vx 

Hence  the  only  complete  system  of  catenaries  which  is  turned  into  a 
complete  system  is  defined  by  the  work  function* 

W=  VX. 

87.  Consider,  for  example,  the  conformal  representation  of  the 
plane 

z  =  x  +  iy  =  rei0 
on  the  plane 

21  =  2?!+  iyi  =  rie'"1 
defined  by 

2l  =   2", 

where  n  is  neither  0  nor  1. 

Here  the  squared  ratio  of  magnification  is 

2(n-l) 

X  =  r2("-1}  =  n"""  . 


*  The  three  physical  cases  mentioned  may  be  included  in  one  general  dis- 
cussion by  considering  the  extremals  of 


J  vmds  =    I  (W  +  h)m2ds  =  minimum; 


when  m  =  1,  we  have  least  action  and  trajectories;  when  m  =  —  1,  least  time 
and  brachistochrones.  For  every  value  of  m  we  obtain,  by  varying  h,  a  sys- 
tem of  oo3  curves.  Cf.  the  general  discussion  of  the  systems  Sk  denned  (for 
arbitrary  fields)  in  Chapter  IV. 


84  THE   PRINCETON  COLLOQUIUM. 

Applying  the  theorems  stated  above,  we  find  that  the  trajectories 
generated  by 


go  over  into  the  trajectories  of  a  new  field 


For  brachistochrones  the  corresponding  fields  are 

W=r-2(n-»}  J^^-Kn-1). 

and  for  catenaries 

W  =  r"-1,         W\=  rr"1. 

The  particular  transformation  z\  =  z2,  that  is,  n  =  2,  gives 
a-ise  to  simple  fields.  Stating  the  results  in  terms  of  the  law  of 
the  central  forces  obtained,  instead  of  the  corresponding  work 
functions,  we  have: 

The  trajectories  of  a  central  force  varying  as  r  (that  is,  the 
conies  described  about  the  center  of  force  as  center)  become 
the  trajectories  of  a  central  force  varying  as  rf~2  (that  is,  the 
conies  described  about  the  center  of  force  as  focus). 

The  brachistochrones  of  a  central  force  varying  as  r~z  become 
the  brachistochrones  of  a  central  force  of  constant  intensity. 

The  catenaries  of  a  central  force  of  constant  intensity  become 
the  catenaries  of  a  central  force  varying  as  r{~3'-. 

88.  Returning  to  the  general  conformal  representation,  we 
observe  that  oo  l  natural  families  forming  a  complete  system  of 
trajectories  can  never  become  a  complete  system  of  brachis- 
tochrones. For  the  trajectories  on  S  due  to  W  +  h  become  oo  l 
natural  families  on  Si,  which,  when  regarded  as  brachistochrones, 
are  due  to  \f(W  +  A);  and  there  is  no  work  function  which 
reduces  this  expression  to  the  form  of  a  function  of  u,  v  plus  a 
constant  depending  only  on  h  .  Thus  for  a  given  (non-homothetic) 
conformal  transformation  there  is  one  system  of  trajectories 


ASPECTS   OF   DYNAMICS.  85 

which  is  converted  into  a  system  of  trajectories,  and  one  system 
of  brachistochrones  which  is  converted  into  a  system  of  brachis- 
tochrones,  but  there  is  no  system  of  trajectories  which  is  converted 
into  a  system  of  brachistochrones.  The  same  is  true  for  any 
two  of  the  three  types  trajectories,  brachistochrones,  catenaries 
or  of  the  infinite  number  of  types  described  in  the  preceding 
footnote  (page  83). 

89.  As  another  application,  consider  the  velocity  curves  con- 
nected with  a  plane  field  of  force  whose  work  function  is  W(x,  y)  . 
For  a  given  speed  v0,  we  obtain  <x>2  such  curves,  defined  by 
the  property  that  the  curvature  at  each  point  and  direction 
equals  the  curvature  of  a  free  particle  starting  out  from  that 
point  and  direction  with  the  speed  v0.  The  differential  equation 
of  this  velocity  system  is 


This  is  recognized  as  a  natural  family;  it  corresponds  to  the  geo- 
desies of  the  surface  whose  first  fundamental  form  is 


By  varying  VQ  we  obtain  the  oc1  velocity  systems  belonging  to 
the  given  field  ;  they  are  pictured  by  the  geodesies  of  oo  :  surfaces. 
Consider  now  a  conformal   representation   of  the  xy-p\ane 
upon  itself.     This  converts  dx-  +  dy-  into 


where  H(x,  y),  by  known  theory,  is  a  harmonic  function.  We 
thus  obtain  oo  *  new  natural  families  corresponding  to  the  geo- 
desies of  the  oo  x  surfaces 


86  THE   PRINCETON  COLLOQUIUM. 

These  <x>  l  natural  families  cannot  usually  be  regarded  as  related 
velocity  systems  for  some  new  field:  the  requisite  condition  is 
that  W  shall  be  the  same  as  //  except  for  a  constant  factor. 

Hence  for  a  given  conformal  transformation  of  the  plane 
(which  is  not  merely  a  similitude),  there  is  a  unique  complete 
velocity  system  belonging  to  a  conservative  field  of  force  which 
is  converted  into  a  complete  system.  The  unique  work  function 
is 

W  =  H  =  log  X, 

where  X  denotes  the  squared  ratio  of  magnifaction  in  the  given 
conformal  representation.  The  fields  obtained  are  Laplacian, 
that  is,  satisfy  the  condition 

Wxx  +  Wn  =  0. 

As  an  example,  the  transformation  Zi  =  log  z  converts  the 
oo  3  velocity  curves  of  the  field  W  =  log  r  (in  which  the  force 
varies  inversely  as  the  distance  from  the  origin)  into  the  <x>3 
velocity  curves  of  the  field  W\  =  Xi  (force  vertical  and  constant)  . 

90.  It  was  shown  above  that  conformal  transformations  are  the 
only  point  transformations  which  convert  every  natural  family 
into  a  natural  family.  Natural  families  are  characterized  by 
properties  A  and  B  of  §  31.  It  is  of  interest  to  notice  that 
property  A  by  itself  is  conformally  invariant.  The  most  general 
system  having  this  property  (that  osculating  circles  constructed 
at  any  point  have  another  point  in  common)  is  what  we  have 
termed  a  velocity  system.  We  now  prove  that 

The  only  point  transformations  which  convert  every  velocity 
system  into  a  velocity  system  are  the  conformal  transformations. 

Consider,  say  the  three-dimensional  case,  where  the  general 
velocity  system  is 

y'2  +  A      *"  -  fc  -  *V)U  +  /  +  A 


The  only  curves  which  are  common  to  all  such  svstems  must 


ASPECTS   OF  DYNAMICS.  87 

satisfy 

1  +  /  +  z'2  =  0,        y"  =  0,        z"  =  0, 

and  are  therefore  the  minimal  straight  lines  of  space.  Since 
the  only  transformations  converting  minimal  lines  into  minimal 
lines  are  conformal,  we  have  the  result  stated.  That  conformal 
transformations  actually  leave  the  velocity  type  invariant  is  easily 
verified  analytically*.  The  result  is  obvious  synthetically  (in 
the  case  of  more  than  two  dimensions)  since  the  conformal  group 
converts  circles  into  circles  and  bundles  of  circles  into  bundles. 
Hence  if  the  original  system  possesses  property  A,  the  same  will 
be  true  of  the  transformed  system. 

91.  It  may  be  shown  that,  for  any  given  non-conformal  trans- 
formation, there  exists  one  and  only  one  velocity  system  which 
is  converted  into  a  velocity  system. 

§§  92-94.    CONTACT  TRANSFORMATIONS 

92.  With  each  natural  family,  or,  what  is  the  same,  with  each 
isotropic  medium,  there  is  associated  a  definite  infinitesimal 
contact  transformation.     This  connection,  which  appears  im- 
plicitly in  Hamilton's  fundamental  memoir  of  1835,  was  worked 
out  in  detail  by  S.  Lie.f 

If  the  index  of  refraction  is  v(x,  y,  z),  the  associated  contact 
transformation  has  the  characteristic  function 


(1)  v(x,  y,  z)  Vl  +  p2+92, 

where  x,  y,  z,  p,  q  are  considered  as  the  coordinates  of  a  surface 
element.  If  the  one-parameter  group  generated  is  applied  to 
an  arbitrary  surface  the  resulting  oo1  surfaces  form  a  wave  set. 
The  trajectories  or  rays  appear  as  the  path  curves  of  this  group. 
Lie  showred  that  the  category  of  transformations  which  thus 

*  Cf.  American  Journal  of  Mathematics,  vol.  27  (1906),  p.  213,  for  the  two- 
dimensional  case. 

t  "  Die  infinitesimalen  Beriihrungstransformationen  der  Mechanik,"  Leip- 
ziger  Berichte  (1889),  pp.  145-153.  A  very  elegant  discussion,  with  new  results, 
is  given  by  Vessiot,  Bull.  Soc.  math,  de  France,  vol.  34  (1906),  pp.  230-269. 


88  THE  PRINCETON   COLLOQUIUM. 

appears,  with  a  characteristic  function  of  type  (1),  and  which  he 
termed  "  the  infinitesimal  contact  transformations  of  mechanics," 
is  distinguished  geometrically  by  the  fact  that  the  so-called* 
transversality  relation  reduces  to  orthogonality. 

93.  The  following  simple  and  easily  proved  theorem  appears 
to  be  new. 

The  alternant  (or  Klammerausdruck  of  Lie)  of  the  contact  trans- 
formations associated  with  any  two  media  is  always  a  point  trans- 
formation. 

94.  Here  we  are  dealing  with  two  natural  families  in  the  same 
three-dimensional  space.     In  connection  with  the  most  general 
problem  of  dynamics  (page  70),  spaces  of  any  dimensionality  must 
be  considered,  with   arbitrary  variable  curvature.     The  space 
depends  on  the  quadratic  form  defining  the  kinetic  energy: 
this  determines  the  quadratic  expression  appearing  under  the 
radical  in  the  generalization  of  (1).     The  potential!  determines 
the  factor  v  which  may  be  any  point  function.     The  general 
theorem  is  then  as  follows: 

The  alternant  of  the  contact  transformations  associated  with  two 
dynamical  problems  (or  natural  families}  will  be  a  point  transforma- 
tion when,  and  only  when,  the  two  expressions  for  the  kinetic  energy 
are  either  the  same  or  differ  by  a  factor  (which  may  be  any  point 
function) ;  the  two  potential  energie^  remain  entirely  arbitrary. 

In  particular,  if  any  two  natural  families  are  constructed  in 
the  same  space  (which  space  is  entirely  arbitrary),  the  alternant 
will  be  a  point  transformation. 

For  a  detailed  discussion  of  the  two-dimensional  case,  in- 
cluding a  number  of  converse  results,  the  reader  is  referred  to 
the  author's  paper,  cited  in  the  first  footnote  below. 

*  Lie  does  not  use  this  term.  The  author  borrows  it  from  the  closely 
connected  problem  in  the  calculus  of  variation.  See  "  The  infinitesimal 
contact  transformations  of  mechanics,"  Bull.  Amer.  Math.  Soc.,  vol.  16  (1910), 
pp.  408-412. 

t  Here  considered  as  including  the  energy  constant  h,  which  is  fixed,  since 
we  are  dealing  with  a  natural  family. 


ASPECTS   OF   DYNAMICS.  89 

§§  95-97.    A  GROUP  OF  SPACE-TIME  TRANSFORMATIONS 

95.  In  the  fundamental  transformation  of  the  relativity  theory, 
known  as  the  Lorentz  transformation,  the  position  coordinates 
x,  y,  z  and  the  time  coordinate  t  are  merged:  the  new  position 
and  the  new  time  appear  as  functions  of  both  the  original  position 
and  the  original  time.  The  Lorentz  group  is  composed  of  the 
linear  transformations  of  the  four  variables  x,  y,  z,  t  which  leave 
invariant  the  quadric 

ar2  +  2/2  +  z2  ~  c2*2  =  0. 

Its  importance  is  due  to  the  fact  that  it  leaves  unaltered  the  form 
of  the  Maxwell  equations. 

We  consider  in  this  section  an  entirely  different  group  of  space- 
time  transformations,  depending  on  arbitrary  functions  instead 
of  arbitrary  constants.  It  arises  in  connection  with  ordinary 
(newtonian)  dynamics  in  the  theory  of  forces  depending  on  the 
time  as  well  as  position. 

We  confine  the  discussion  for  the  sake  of  simplicity  to  the  case 
of  two  dimensions.  What  transformations  of  the  three  vari- 
ables x,  y,  t  will  convert  any  set  of  equations  of  the  form 

ds*c  d  u 

(1)  -ftp  =  <r>(x,  y,  t),      jp  =  t(x,  y,  t) 

into  another  set  of  the  same  form?  An  arbitrary  transformation 
would  produce  equations  representing*  a  force  depending,  not 
only  on  x,  y,  t,  but  also  on  the  velocity  dx/dt,  dy/dt.  The  problem 
is  to  find  those  peculiar  transformations  which  do  not  introduce 
the  velocity  in  the  final  equations.  The  result  is  as  follows : 

The  only  space-time  transformations  which  convert  every  space- 
time  field  of  force  into  a  space-time  field  are  those  of  the  form 


(2)  t,  =  f(t),        x,  =  (ax  +  by)  4f'(t)  +  0(0, 


The  group  thus  involves  three  arbitrary  functions  f(t),  g(t],  h(t)  as 
well  as  four  arbitrary  constants  a,  b,  c,  d. 


90  THE   PRINCETON   COLLOQUIUM. 

96.  Another  representation  of  the  same  group,  which  has  the 
advantage  of  avoiding  radicals,  is 


-n  =  (ax 

(3) 

y,  =  (ex  +  dt)\(t)  +  v(t}. 

When  such  a  transformation  is  applied  to  equations  (1), 
the  new  equations  are  found  to  be 

X^  =  (XX  -  2X2)(aa:  +  by)  -f  X2(a^  +  6</0  +  X/i  -  2Xju, 
X5#!  =  (XX  -  2X2)(cz  +  dy)  +  X2(c^  +  <ty)  +  \v  -  2\v. 

Of  course  the  original  variables  x,  y,  t  are  here  to  be  replaced  by 
their  values  in  the  new  variables  x\,  y\,  t\. 

97.  The  transformation  converts  the  space-time  curves  of  the 
original  force  into  the  space-time  curves  of  a  new  force.  Of 
course  it  is  not  a  point  transformation  of  the  :n/-plane,  so  it  does 
not,  as  was  the  case  for  the  Appell  transformation  (page  76), 
convert  trajectories  into  trajectories.  These  remarks  apply  even 
in  the  special  case  where  the  force  is  positional.  Consider,  as 
a  simple  example,  the  transformation 

ti  =  \eu,        xi  =  xel,    2/1  =  ye1, 
applied  to  the  equations 

x  =  x,       y  =  y. 

The  transformed  equations  are  found  to  be 
£1  =  0,        i/i  =  0. 

The  first  field  is  central,  the  force  varying  directly  as  the  distance, 
so  that  the  trajectories  are  oo3  conies  with  the  same  center. 
The  second  force  is  everywhere  zero,  so  the  trajectories  are 
merely  oo  2  straight  lines. 


CHAPTER   IV 

CONSTRAINED  MOTIONS  IN  A  FIELD.       GENERALIZATION  OF 

THE  TRAJECTORY  PROBLEM  INCLUDING  BRACHIS- 

TOCHRONES  AND  CATENARIES 

§§  98-114.    SYSTEMS  Sk  DEFINED  BY  P  =  kN 

98.  In  connection  with  a  field  of  force,  the  only  curves  usually 
studied  are  the  lines  of  force  and  the  trajectories.     In  the  plane 
the  lines  of  force  form  a  simply  infinite  system,  and  the  tra- 
jectories a  triply  infinite  system.     The  former  system  has  no 
peculiar  properties,  since  any  set  of  <x> l  curves  may  be  regarded 
as  the  lines  of  force  in  some  field,  in  fact  in  an  infinite  number  of 
•different  fields.     The  triply  infinite  system  of  trajectories  has 
peculiar  properties  which  have  been  discussed  in  Chapter  I. 
Other  noteworthy  systems  of  curves  are  connected  with  the  field, 
for  example,  brachistrochrones,  catenaries,  velocity  curves,  and 
tautochrones. 

99.  Omitting  the  tautochrones,  the  other  three  systems  named, 
together  with  the  trajectories,  may  all  be  obtained  as  special  cases 
of  this  simple  general  problem :  to  find  curves  along  which  a  con- 
strained motion  is  possible  such  that  the  pressure  is  proportional 
to  the  normal  component  of  the  force. 

100.  If  an  arbitrary  curve  is  drawn  in  the  plane  field  of  force, 
and  the  particle,  of  say  unit  mass,  is  started  along  it  from  one 
of  its  points  with  a  given  speed,  the  constrained  motion  along 
the  given  curve  is  determined.     The  acceleration  along  the  curve 
is  given  by  T,  the  tangential  component  of  the  force  vector. 
So  the  speed  at  any  point  is  determined  by 

(1)  tf  =  fids. 

The  pressure  P  (of  course  normal  to  the  curve,  since  the  curve 

91 


92  *  THE   PRINCETON    COLLOQUIUM. 

is  considered  smooth)  is  given  by  the  elementary  formula 
(2)  P=*-N. 

If  we  increase  the  initial  speed,  the  effect  is  to  increase  v2  by  a 
constant  tf;  and  hence  P  changes  by  the  addition  of  a  term  of  the 
form  cfr. 

101.  If  the  given  curve  is  a  trajectory,  the  initial  speed  may  be 
so  chosen  that  the  pressure  vanishes  throughout  the  motion; 
that  is,  trajectories  may  be  defined  as  curves  of  no  constraint. 
Of  course,  if  a  different  initial  speed  is  used,  P  will  be  of  the  form 
cfr;  but,  as  regards  the  curves,  they  are  completely  characterized 
by  P  =  0. 

102.  If  the  given  curve  is  a  brachistochrone  and  if  the  motion 
along  it  is  brachistochronous,  Euler  proved  (assuming  the  force 
to  be  conservative)  that  the  pressure  was  double  the  normal 
component  of  the  acting  force  and  opposite  to  it  in  direction, 
that  is,  P  =  —  2N.     If  the  force  is  not  conservative,  the  real 
brachistochrones,  as  defined  by  a  problem  of  the  calculus  of  varia- 
tions, form  a  quadruply  infinite  system.     The  curves  defined  by 
the  property  P  =  —  2  AT  then  form  a  triply  infinite  system  of  what 
should   be   called   pseudo-brachistochrones.     These   curves   are 
really    brachistochrones    only    in    the    conservative    case.     No 
ambiguity  however  will  arise  by  terming  the  system  here  con- 
sidered brachistochrones  instead  of  pseudo-brachistochrones. 

103.  The  general  problem  suggested  is  to  find  curves  such  that  P 
shall  be  proportional  to  N.     So  P  =  kN.     To  a  given  value  of 
k  there  correspond  =o3  such  curves:  the  system  so  obtained  will 
be  denoted  by  Sk-     The  four  special  cases  of  physical  interest  are 
as  follows: 

k  =  0  gives  So,  the  system  of  trajectories; 

k  =  —  2  gives  S-%,  the  system  of  brachistochrones; 

k  =  1  gives  Si,  the  system  of  catenaries; 

k  =  oo  gives  S^,  the  system  of  velocity  curves. 


ASPECTS   OF   DYNAMICS.  93 

104.  The  last  case  requires  a  justification  in  terms  of  limits 
which  is  easily  carried  out  analytically. 

105.  The  third  case  follows  from  the  known  fact  that  when  an 
inextensible  flexible  homogeneous  string  is  suspended   in  any 
field  of  force,  the  resulting  form  of  equilibrium,  called  a  catenary 
in  the  general  sense  of  the  term,  has  the  dynamical  property 
that  when  a  particle,  started  out  with  the  proper  initial  velocity, 
rolls  along  the  curve,  the  pressure  at  any  point  equals  the  normal 
component  of  the  force:  that  is,  catenaries  are  defined  by  P  =  N, 
corresponding  to  k  =  1. 

106.  Of  course  a  triply  infinite  system  Sk  exists  for  any  value 
of  the  parameter  k.     The  differential  equation  of  the  system, 
in  intrinsic  form,  is  easily  obtained  by  eliminating  v  from  the 
equations 

(3)  v*/r  =  (k  +  1)N,        vvs  =  T. 
The  result  is 

(4)  Nrs  =  (n+  1)T  -  rW, 
where 

(4')  n 


We  may  readily  find  various  properties  from  this  intrinsic 
equation,  but  in  order  to  obtain  a  complete  set  it  is  necessary  to 
have  recourse  to  the  equivalent  equation  in  cartesian  coordinates 

(*  -  y  W"  =  {*.  +  Gh,  -  v*W 


-     3  +  -  -i-2    -  \y ". 

1  +  r 

This  obviously  reduces  to  the  familiar  trajectory  equation  of  §1 
when  n  —  2,  corresponding  to  k  =  0.  Brachistochrones  cor- 
respond to  n  =  —  2,  catenaries  to  n  =  1,  velocity  curves  to 
n  =  0. 

107.  We  now  state  the  characteristic  properties  of  a  system  of 
the  above  type  for  any  value  of  n,  that  is,  any  value  of  k. 


94  THE  PRINCETON   COLLOQUIUM. 

Characteristic  Properties  of  the  System  Sk 

Property  1. — For  any  given  element  (x,  y,  y')  the  foci  of  the 
osculating  parabolas  of  the  single  infinity  of  curves  determined 
by  the  given  element  lie  on  a  circle  passing  through  the  given  point. 

Property  2. — At  any  point  0  the  tangent  of  the  angle  which  the 
focal  circle  makes  with  the  given  element  is  to  the  tangent  of 
the  angle  which  the  given  element  makes  with  a  certain  direction 
fixed  at  0  (the  direction  of  the  acting  force)  as  3  is  to  n  +  1, 
that  is,  as  3k  +  3  is  to  k  +  3. 

Property  8.  Through  a  given  point  there  pass  a  single  infinity 
of  curves  admitting  hyperosculating  circles  of  curvature;  the 
centers  of  these  circles  lie  on  a  conic  passing  through  the  given 
point  in  the  direction  of  the  force  vector. 

Property  4- — The  normal  at  the  given  point  0  cuts  the  conic 
described  in  property  3,  at  a  distance  equal  to  n  +  1,  that  is 
(k  +  3)/(fc  +  1),  times  the  radius  of  curvature  of  the  line  of 
force  passing  through  0. 

Property  5. — This  is  of  the  same  form  as  property  V  ( §  3) 
obtained  in  the  discussion  of  trajectories,  the  number  3  being 
replaced  by  the  number  n  +  1.  In  the  notation  of  page  11 

d  1  j?._l COCOsy   —    COzWy      _ 

dxAA'^~  dyBB'^     (n+  l)co2 

108.  The  special  case  where  n  equals  —  1,  that  is,  the  system 
8-3,  is  exceptional  and  requires  a  separate  discussion;  but  as 
we  do  not  need  the  results,  this  case  is  omitted. 

109.  While  the  properties  corresponding  to  different  values 
of  k  are  analogous,  they  are  of  course  not  identical.     The  first 
property  is  common  to  all  the  systems.     But  the  second  property 
involves  the  parameter  k.     Thus,  while  for  trajectories  the  con- 
stant ratio  that  appears  is  1  (bisection),  it  is  —  3  for  brachisto- 
chrones,  3/2  for  catenaries,  and  3  for  velocity  curves.     Not  only 
are  the  triply  infinite  systems  Sk,  corresponding  to  different 
values  of  k,  distinct  in  any  given  field  of  force,  but  also  no  two 


ASPECTS   OF  DYNAMICS.  95 

systems  arising  in  two  distinct  fields  can  ever  coincide.  For 
example,  if  a  certain  system  of  °o3  curves  arises  as  trajectories 
in  one  field,  it  cannot  also  arise  as  catenaries  in  either  the  same 
or  another  field. 

110.  If  we  combine  all  the  systems  Sk,  in  a  given  field  (<p,  ^), 
we  obtain  a  quadruply  infinite  system  which  we  now  proceed 
to  study.  The  differential  equation  of  the  fourth  order  defining 
this  system  is  readily  obtained  by  eliminating  k  from  the  equation 
of  Sk.  It  is  more  convenient  to  carry  this  out  in  terms  of  in- 
trinsic quantities,  using  either  the  radius  of  curvature  and  its 
first  and  second  derivatives  with  respect  to  the  arc,  quantities 
denoted  by  r,  rs,  ras,  or  else  the  radius  of  curvature  together  with 
the  radii  of  the  first  and  second  evolute,  quantities  which  we 
denote  by  r,  r\,  rz.  The  two  sets  of  quantities  are  equivalent, 
being  connected  by  the  relations  rt  =  rrs,  r»  =  r2rss  +  rr«2.  The 
equation  of  the  quadruply  infinite  system  may  then  be  put,  using 
the  notation  of  §  2,  into  the  form 

Nr,  +  rft  -       0  = 


This  may  be  written  in  either  of  the  forms 


where  the  /8's  are  functions  of  x,  y,  y'. 

111.  We  notice  first  that  r2  is  quadratic  with  respect  to  r\. 
Hence  for  given  values  of  x,  y,  y',  r,  that  is  for  a  given  curvature 
element,  the  °o  1  curves  of  the  system  have  the  property  that  the 
locus  of  the  third  center  of  curvature  is  a  parabola  with  axis 
parallel  to  the  fixed  radius  of  curvature,  that  is,  perpendicular 
to  the  initial  direction  y'. 

112.  An  equivalent  statement  is  this:  If  for  each  of  the  curves 
we  construct  the  osculating  conic  (five-point  contact),  the  locus 


yb  THE   PRINCETON   COLLOQUIUM. 

of  the  centers  of  these  conies  is  a  conic  passing  through  a  given 
point  in  the  given  direction.  It  is  perhaps  worth  while  to  restate 
this,  so  far  as  it  concerns  the  four  special  cases  of  physical  interest, 
as  follows:  In  any  plane  field  of  force  select  any  fixed  element  of 
curvature;  corresponding  to  the  initial  values  of  x,  y,  yr  and  r  so 
given,  construct  the  unique  trajectory,  unique  brachistochrone, 
unique  catenary,  the  unique  velocity  curve,  and  the  respective 
centers  of  the  osculating  conies  ;  the  four  centers  so  found  and 
the  given  point  (a:,  y}  will  lie  on  a  conic  passing  through  the  latter 
point  in  the  given  direction  y'.  (Cf.  the  first  footnote  on  page  98.) 

113.  Keeping  the  curvature  element  fixed  and  varying  the 
parameter  k,  the  value  of  rs  or,  what  is  equivalent,  of  r\,  varies 
linearly.     As  above,  let  n  denote  the  fraction  2f(k  +  1)  ;  then  if 
values  of  n  forming  an  arithmetic  progression  are  selected,  the 
corresponding  values  of  r\  also  form  an  arithmetic  progression. 
The  successive  differences  in  the  values  of  r\  corresponding  to 
the  case  of  trajectory,  brachistochrone,  catenary,  and  velocity 
curve  are  proportional  to  4,  —  3,  1. 

114.  If  in  the  system  Sk  we  keep  x,  y,  y'  fixed  and  vary  r,  two 
limiting  cases  of  interest  arise.     First,  if  r  becomes  infinite,  then 
r8  is  also  infinite,  and  the  limiting  curve  obtained  is  a  straight 
line.     In  fact  the  <x>2  straight  lines  of  the  plane  form  part  of 
every  system  Sk. 

On  the  other  hand,  if  r  approaches  zero,  then  ra  approaches  a 
definite  limit 


Remembering  that  the  tangent  of  the  angle  of  deviation  is  one 
third  of  rt,  we  may  state  the  result  obtained  as  follows:  In  any 
system  Sk  if  we  take  any  lineal  element  and  let  r  approach  zero, 
the  tangent  of  the  corresponding  angle  of  deviation  is  to  the 
tangent  of  the  angle  which  the  force  vector  makes  with  the  normal 
to  the  given  element  in  the  fixed  ratio  of  n  -f  1  to  3.  The  special 
values  of  this  ratio  for  the  four  special  systems  of  physical 
interest  are  respectively  1,  —  1/3,  2/3,  1/3.  In  the  case  of  tra- 
jectories, it  is  noteworthy  that  the  limiting  position  of  the  axis 


ASPECTS   OF   DYNAMICS.  97 

of  deviation  coincides  with  the  direction  of  the  force  acting  at 
the  given  point. 

§§  115-116.    CURVES  OF  CONSTANT  PRESSURE 

115.  We  now  consider  a  second  simple  generalization  of  the 
problem  P  =  0,  defining  trajectories.  We  consider,  namely, 
curves  corresponding  to  P  =  c,  where  c  denotes  any  constant. 
The  curves  obtained  may  be  termed  curves  of  constant  pressure  : 
only  along  such  a  curve  is  a  constrained  motion  of  a  particle 
possible  such  that  the  pressure  against  the  curve  remains  constant. 

For  a  given  value  of  c  a  system  of  <x>3  such  curves  is  obtained, 
whose  intrinsic  equation,  found  by  differentiating  the  relation 


P  =  tf/r  -  N  =  c, 
is 

(c+  N)ra  =  3T-  9t. 

We  see  that  this  system  for  any  value  of  c  retains  property  I  of 
the  system  of  trajectories.  Omitting  the  discussion  of  the  higher 
properties  of  these  triply  infinite  systems  we  consider  the  quad- 
ruply  infinite  system  whose  differential  equation,  found  by  elimi- 
nating c,  may  be  written  in  either  of  the  intrinsic  forms 


-  37V)r..  =  (2r9i  -  7>s2  +  [9V  +  (9?2  -  3£)r  -  3N]rs, 

-  37>2  =  (3r9t  -  4Z>i2  +  [9V  +  (9t2 


This  gives  the  totality  of  oo  4  curves  of  constant  pressure  defined 
by  a  given  field. 

As  regards  special  cases  of  interest,  we  note,  in  addition  to 
c  =  0,  giving  trajectories,  the  case  c  =  QO  which  gives  rs  =  0, 
defining  circles;  hence  for  any  field  of  force  the  oo4  curves  of 
constant  pressure  include  the  oc  3  circles  of  the  plane,  which  arise 
in  fact  as  curves  of  infinite  pressure. 

116.  The  quadruply  infinite  system  which  here  arises,  as  wrell 
as  that  obtained  in  the  previous  problem  P  =  kN,  comes  under 
15 


98  THE  PRINCETON  COLLOQUIUM. 

the  category  represented  by  a  differential  equation  of  the  type* 
f  =  Ay'"2  +  By'"  +  C. 

It  therefore  enjoys  the  property,  previously  stated  in  the  other 
problem  (§  112),  that  the  locus  of  the  centers  of  the  osculating 
conies  corresponding  to  any  element  (x,  y,  y',  y")  is  a  conic 
touching  the  element  (x,  y,  y'}.  Of  course,  since  the  forms  of 
A,  B,  C  in  the  two  problems  are  quite  distinct,  the  systems  are 
distinguished  in  their  higher  properties. 

§§  117-118.    TAUTOCHRONES 

117.  Tautochrones  are  not  included  in  either  of  the  previous 
problems.  They  are  not  distinguished  by  any  simple  law  of  pres- 
sure, f  The  condition  for  a  tautochrone  is  that  the  resulting  con- 
strained motion  of  a  particle  along  the  curve  be  harmonic,  that  is, 

(1)  T  =  k(s  -  «0), 

where  k  is  a  constant  (which  is  negative  for  actual  and  positive 
for  virtual  tautochrones)  and  s  —  So  denotes  the  arc  reckoned 
from  a  fixed  point  of  the  curve,  the  center  of  the  tautochronous 
motion.  From  this 

(2)  Tu  =  0 

and  hence,  by  expansion,  the  general  equation  of  the  system  of  <x>3 
tautochrones  in  any  field  is\ 

(3)  tfr.-^f +(£,  +  5R)r-  T, 

where  the  notation  is  that  of  §  2. 

*  This  type  (noteworthy  in  that  it  unifies  many  distinct  mathematical  and 
physical  problems)  first  presented  itself  in  the  author's  study  of  "Systems 
of  extremals  in  the  calculus  of  variations,"  Bull.  Amer.  Math.  Soc.,  vol.  13 

(1907),  p.  290:  the  extremals  of  any  integral  of  the  second  order  J  f(x,  y,  y',  y")dx 
form  a  system  of  that  type.     In  these  lectures  other  physical  problems  leading 
to  species  included  in  this  type  are  treated  in  §§  110,  135,  137. 
t  It  may  be  shown  that  during  any  tautochronous  motion 

P  =  k(s  -  s0)2/r  -  N. 

t  "  Tautochrones  and  brachistochrones,"  Bull.  Amer.  Math.  Soc.,  vol.  15 
(1909),  pp.  475-483. 


ASPECTS   OF  DYNAMICS.  99 

We  see  that  r«  is  a  quadratic  function  of  r,  and  not  a  linear 
function  as  in  the  case  of  trajectories  and  the  other  systems  Sk- 
For  a  discussion  of  the  geometric  properties  of  tautochrones,  we 
refer  to  the  dissertation  of  H.  W.  Reddick.* 

118.  There  is  no  field  in  which  the  tautochrones  coincide  with 
the  trajectories,  or  with  any  of  the  systems  Sk,  in  either  the  same 
or  some  other  field,  except  for  the  case  k  =  —  2  corresponding 
to  brachistochrones.  The  classical  work  of  Huygens  and  J. 
Bernoulli  showed  that  for  a  uniform  field  the  system  of  tauto- 
chrones is  identical  with  the  system  of  brachistochrones.  The 
author  has  shown  that  the  only  other  field  where  such  duplication 
occurs  is  that  in  which  the  force  is  central  and  varies  directly 
as  the  distance.  The  only  case  of  duplication  in  two  distinct 
fields  is  as  follows:  The  tautochrones  of  the  field  <p  =  0,  \f/  =  y 
coincide  with  the  brachistochrones  of  the  field  <p  =  0,  ^  =  y~3. 
The  particular  fields  arising  in  this  duplication  problem  are  in- 
cluded in  the  interesting  class  of  fields,  involving  eight  parameters, 
characterized  by  the  vanishing  of  the  element  function  T\. 
For  such  a  field  rg,  according  to  (3),  becomes  linear  in  r,  and  hence 
the  oo 2  straight  lines  of  the  plane  are  included  in  the  system  of 
tautochrones.  f 

118'.  Each  of  the  «>3  tautochrones  in  a  given  field  has  asso- 
ciated* with  it  a  certain  time  of  oscillation,  determined  by  the 
value  of  the  constant  &  in  (1).  To  each  value  of  the  period,  that 
is,  to  each  value  of  k,  corresponds  a  certain  family  of  oo 2  tauto- 
chrones, whose  differential  equation,  in  implicit  form,  is 

r(k-  £)  ='N, 
or,  expanded, 


We  pass  over  the  easy  geometric  interpretation ;  and  note  merely 
the  special  family,  corresponding  to  the  value  k  =  0,  for  which 

*  Amer.  Jour,  of  Math.,  vol.  33  (1911). 

t  The  corresponding  problem  in  space  is  treated  in  Reddick's  paper  and 
gives  a  class  of  fields  involving  twenty  parameters. 


100  THE   PRINCETON   COLLOQUIUM. 

the  period  is  infinite.     This  separates  the  actual  from  the  virtual 
tautochrones. 

§  119.    NON-UNIFORM  CATENARIES 

119.  It  is  a  familiar  fact  that  vertical  parabolas  appear  in 
elementary  dynamics  in  two  distinct  discussions;  first,  as  trajec- 
tories of  a  cannon  ball,  and  secondly  as  forms  of  equilibrium  of 
a  chain  in  which  the  mass  (or  load)  of  any  element  is  propor- 
tional to  the  horizontal  projection  of  that  element.  Here  the 
force  is  ordinary  gravity.  The  question  arises  whether  any  other 
fields  of  force  give  rise  to  a  like  duplication. 

We  first  consider  the  following  general  problem  of  non-uniform 
catenaries.  If  a  flexible  string  or  chain,  in  wrhich  the  mass  of 
any  element  of  length  is  proportional  to  some  given  function  p  of 
x,  y,  y',  is  suspended  in  a  positional  field,  the  possible  forms  of 
equilibrium  are  defined  by  the  equation 

This  represents  the  <x> 3  non-uniform  catenaries  for  a  given  field 
<p(xy),  \f/(xy~)  and  a  given  density  law  n(x,  y,  y'),  where  M  denotes 

log/i. 

On  the  other  hand,  the  trajectories  in  the  given  field  are  (Defined 
by  the  equation 

Nr,=  371-  r9J. 

Our  problem  then  is  to  find  those  fields  for  which  the  two 
systems  described  coincide.  The  result  obtained  is  that  the  field 
must  be  central  or  parallel.  The  detailed  result  is  as  follows: 

In  any  central  field  of  force  the  °c3  trajectories  may  be  also 
obtained  as  catenaries  by  loading  the  chain  so  that  its  density  is 
proportional  to  the  perpendicular  dropped  from  the  center  to 
the  tangent  line.  In  the  more  special  case  where  the  field  is 
parallel,  the  density  is  proportional  to  the  sine  of  the  angle 
between  the  element  of  the  curve  and  the  force. 


ASPECTS   OF  DYNAMICS.  101 

It  is  easy  to  obtain  analogous  comparisons  between  brachisto- 
chrones  and  catenaries.  In  this  case  the  density  must  vary 
inversely  as  the  cube  of  the  perpendicular  dropped  from  the 
center  (or  of  the  sine  of  the  angle  referred  to  above).  For 
example,  in  the  case  of  gravity  the  vertical  cycloids  which  appear 
as  brachistochrones  may  be  obtained  as  catenaries  by  causing 
the  load  applied  to  any  element  to  vary  inversely  as  the  cube  of 
its  horizontal  projection. 

All  the  results  may  be  included  in  a  generalization  found  by 
comparing  the  non-uniform  catenaries  with  the  systems  denoted 
by  Sk  in  §  103.  The  density  must  vary  as  the  (n  —  l)th  power 
of  the  perpendicular,  where  n  is  the  number  defined  on  page  93. 
The  field  is  necessarily  central  or  parallel. 


CHAPTER  V 

MORE  COMPLICATED  TYPES  OF  FORCE 

§§  120-122.    MOTION  IN  A  RESISTING  MEDIUM 

120.  We  consider  the  motion  of  a  particle  moving  in  the  plane 
under  a  positional  field  of  force  and  influenced  by  a  resisting 
medium,  the  resistance  acting  in  the  direction  of  the  motion  and 
varying  as  some  function  of  the  speed  v.  The  equations  of 
motion  will  then  be  of  the  form 

(1)  x  =  <p(x,  y)  +  xf(v),        y  =  t(x,  y}  +  yf(v), 

where  the  resistance  R  is  equal  to 

R  =  vf(v). 
The  differential  equation  of  the  trajectories  is  found  to  be 

(*  -  y  W"  =  {**  +  y'(t*  -  *>«)  -  A>»l 


where  the  argument  v  of  /  is  to  be  expressed  in  terms  of  x,  y, 
y't  y"  by  means  of 


y" 

Consider  now  the  °° l  trajectories  starting  from  a  given  element 
(xt  y,  y'}.  The  focal  locus,  that  is,  the  locus  of  the  foci  of  the 
osculating  parabolas,  varies  in  shape  with  the  function  /,  that 
is,  with  the  law  of  resistance. 

WTe  know  that,  if  there  is  no  resistance,  property  I  of  §  3  holds, 
that  is,  the  focal  locus  is  a  circle  passing  through  the  given  point. 
Are  there  any  resisting  media  for  which  this  property  is  pre- 
served? A  simple  discussion  shows  that  there  are,  the  appro- 

102 


ASPECTS   OF  DYNAMICS.  103 

priate  media  being  those  for  which  R  is  of  the  form  Av2  -f-  B. 

For  such  media,  property  II  will  not  usually  be  fulfilled;  in 
fact  the  only  medium  preserving  the  properties  I  and  II  is  that 
in  which  the  resistance  varies  as  the  square  of  the  speed. 

If  we  impose  also  property  III,  both  A  and  B  must  vanish, 
that  is,  the  resistance  vanishes  and  the  force  is  purely  positional. 

It  is  of  interest  to  examine  the  case  where  the  resistance  varies 
as  any  power  vn  of  the  speed.  The  differential  equation  of  the 
trajectories  is  then  of  the  form 

y'"  =  ay"  +  by"*  +  cy"m, 
where 

m  =  |(4  —  n). 


The  focal  locus  is  a  curve  whose  inverse  with  respect  to  the  given 

point  is 

X=  at+biY 


This  becomes  a  straight  line  (as  in  the  case  of  no  resistance), 
when  m  is  1  or  2,  that  is,  when  n  is  2  or  0. 

The  curve  is  a  conic  when  m  is  3  or  0  or  3/2,  that  is,  when  n  has 
one  of  the  values  —  2  or  4  or  1.  When  n  =  —  2  the  conic  is  a 
parabola  with  its  axis  parallel  to  the  given  element.  When 
n  =  4  it  is  a  hyperbola,  asymptotic  to  the  line  of  the  given  initial 
element.  When  n  =  1  it  is  a  parabola  touching  the  initial  line 
(not  at  the  given  point). 

121.  We  now  state  briefly  the  corresponding  results  in  ordinary 
space.     No  matter  what  the  law  of  resistance  is,  property  I 
(of  the  set  of  four  properties  for  space  given  in  §  11)  is  fulfilled; 
for   the   osculating   planes  necessarily  pass  through  the  force 
vector.     The  only  laws  for  which  property  II  is  preserved  are 

those  included  in 

R  =  Av2  +  B. 

If  property  III  is  also  to  be  preserved,  the  resistance  must  vanish. 

122.  The  results  may  be  derived  easily  from  the  intrinsic 
equations 

(3)  v*  =  rN,        vvs  =  T  +  R, 


104  THE   PRINCETON   COLLOQUIUM. 

obtained  by  taking  components  of  the  acting  forces  along  the 
normal  and  tangent  to  the  trajectory.  The  geometric  equation, 
resulting  from  the  elimination^  v,  is  of  the  form* 

(4)  Nra=  -  r91  +  3r+2fl. 

This  gives  the  relation  between  ra  (the  rate  of  variation  of  r 
with  respect  to  *)  and  r  (the  radius  of  curvature).  The  resistance 
R,  which  is  given  as  a  function  of  v,  is  here  to  be  expressed  in 
terms  of  r  by  means  of  the  first  of  the  relations  (3).  If  prop- 
erty I,  of  plane  trajectories,  is  to  hold,  rs  must  be  a  linear  integ- 
ral function  of  r;  this  will  be  the  case  not  only  when  R  vanishes, 
but  also,  as  stated  above,  when  it  is  of  the  form  AT?  +  B. 

§§  123-126.    PARTICLE  ON  A  SURFACE 
123.  The  motion  of  a  particle  on  any  constraining  surface 

x  =  <p(u,  V),        y  =  t(u,  u),        z  =  x(u,  v) 

under  any  positional  forces  may  be  investigated  most  simply  by 
means  of  the  Lagrangian  equations 


.  _=  _ 

dt\du)  "  di>  ~~       '     dt\dv      "  dv 

where  T  is  the  kinetic  energy 

2T  =  Etf+IFuv+Gi? 

and  U,  V  are  the  components  of  the  force  given  as  functions  of 
u,  fl.f     The  explicit  equations  of  motion  are  of  the  form 


u  = 
v  = 


*  From  this  we  may  obtain  the  following  dynamical  result :  If  a  particle 
starts  from  rest,  the  initial  radius  of  curvature  of  the  trajectory  is  to  the 
radius  of  curvature  of  the  line  of  force  passing  through  the  initial  point  as 
371  +  2R  is  to  T.  When  R  vanishes  we  have  the  simple  result  previously 
stated. 

t  See  for  example  Whittaker,  Analytical  Dynamics,  p.  390,  and  Hada- 
mard,  Jour,  de  Moth.  (5),  vol.  3,  p.  331. 


ASPECTS    OF   DYNAMICS.  105 

where  3>,  ^  define  the  force  and  the  A's  and  B's  are  functions  of 
uy  v  depending  only  on  the  given  surface. 

124.  We  observe  that  here  u,  v  depend  not  only  on  the  position 
u,  v  but  also  upon  the  velocity  u,  v.  Hence  the  motion  in  the 
wfl-plane  corresponding  to  the  actual  motion  on  the  surface 
is  not  usually  generated  by  any  positional  force  in  that  plane. 
The  only  exception  arises  when  the  A's  and  the  B's  vanish 
identically:  this  is  the  case  only  if  the  given  surface  is  develop- 
able, and  if  its  representation  on  the  wt>-plane  differs  from  its 
development  on  the  plane  by  at  most  an  affine  transformation. 

Another  problem  including  this  as  a  special  case  is  to  deter- 
mine when  the  motion  in  the  wi>-plane  can  be  regarded  as  due 
to  a  positional  force  together  with  a  resistance  acting  in  the 
direction  of  the  motion.  The  condition  for  this  is 

A0v?  +  2Aiuv  -f-  A^v2      u 


B0u2  +  2Bluv  +  Bzv2       )  ' 

Expanding,  we  find  four  conditions  on  the  six  functions  A,  B, 
which  turn  out  to  be  precisely  the  conditions  that  the  geodesies 
of  the  surface  shall  be  pictured  by  straight  lines,  a  result  which 
may  be  proved  directly.  Hence  the  only  case  in  which  the 
motion  on  the  surface  is  pictured  in  the  z<»-plane  by  a  motion 
due  to  a  positional  force  together  with  a  resistance  depending  on 
the  velocity  components  and  acting  in  the  direction  of  the  motion, 
is  that  in  which  the  surface  has  constant  curvature  and  the  rep- 
resentation is  geodesic. 

125.  We  proceed  with  the  general  equations  of  motion.  If 
we  eliminate  the  time,  we  obtain  the  differential  equation  of  the 
third  order  defining  the  °o3  trajectories  in  the  form 


=  {50 


/ 


where  the  coefficients  are  functions  of  u,  v.     We  confine  our- 
selves  to   the  observation   that   the  picture  curves  in  the  uv- 


106  THE   PRINCETON  COLLOQUIUM. 

plane  come  under  the  type 


where  the  coefficients  are  lineal-element  functions:  the  focal  locus 
is  thus  not  a  circle,  but  a  special  quartic.  Hence  if  we  consider 
the  oo !  trajectories  on  the  surface  obtained  by  starting  a  particle 
at  a  given  point  in  a  given  direction  with  different  speeds,  the 
picture  curves  in  the  wo-plane  have  osculating  parabolas  at  the 
common  point  whose  foci  lie  on  a  special  quartic  curve. 

126.  What  is  the  simplest  property  of  the  actual  trajectories 
described  on  the  surface?  -What  is,  in  particular,  the  locus  of 
the  osculating  spheres  of  the  oo l  trajectories  considered? 

To  answer  this  we  take  our  surface  not  in -parametric  form,  but 
in  the  explicit  form 

z  =  f(x,  y). 

We  may  take  the  given  point  as  origin,  the  tangent  plane  as  the 
zy-plane,  and  the  fixed  initial  direction  as  that  of  the  axis  of  x. 
We  find,  by  differentiating  the  equation  of  the  surface  and  making 
use  of  y'  =  0,  z'  =  0,  that 

z"  =  a>        z'"  =  b  +  cy", 

where  a,  b,  c  are  constants,  equal  respectively  to  the  values  of 
the  partial  derivatives  fxx,  fxxx,  4fxy  at  the  origin.  Again,  from 
the  general  equation  of  the  trajectories,  we  have  a  relation  of  the 
form 

y'"  =  a  +  ft"  +  yy"\ 

The  center  of  the  osculating  sphere  of  the  trajectory  is  then 

X  =  0, 

z'"          =  b  +  cy" 

=  y"z"'  -  z"y'"  =  y"(b  +  Cy")  -  a(a+py"  +  yy"z) ' 

"  =  „.'/„/"        -.",."'  ~ 


y"z'"  -  z"y'"      y"(b  +  cy")  -  a(a 


ASPECTS   OF   DYNAMICS.  107 

Here  y"  enters  as  parameter,   varying  from  curve  to  curve: 
eliminating  it,  we  find  the  locus,  lying  in  the  plane  X=  0,  to  be 

a72  +  /3F(l-aZ)  +  7(l  -  aZ)z  +  Z{bY  +  c(l  -  aZ)\  =  0. 

Hence  for  any  positional  force  on  any  surface,  the  <x>  l  trajectories 
starting  from  a  given  lineal  element  of  the  surface  have  osculating 
spheres,  at  the  common  point,  whose  centers  lie  on  a  (general)  conic 
in  the  plane  normal  to  the  element. 

This  conic  passes  through  the  center  of  curvature  of  the  normal 
section  of  the  surface  determined  by  the  given  element.  If  the 
element  is  in  one  of  the  principal  directions  of  the  surface,  the 
conic  touches  the  normal  to  the  surface. 

§§  127-130.    THE  GENERAL  FIELD  IN  SPACE  OF  ^-DIMENSIONS 

127.  Any  dynamical  system  with  n  degrees  of  freedom  may  be 
represented  by  a  particle  in  space  of  n  dimensions.  For  example, 
an  arbitrary  rigid  body  in  ordinary  space  is  represented  by  a 
particle  in  six-dimensional  space,  and  the  astronomical  problem 
of  three  bodies  in  the  most  general  case  leads  to  a  representative 
particle  in  space  of  nine  dimensions. 

For  conservative  forces,  or  natural  families,  the  general  dis- 
cussion for  any  dimensionality  has  already  been  given  (§  69). 
We  shall  not  attempt  a  complete  discussion  for  arbitrary  posi- 
tional forces  (corresponding  to  that  given  in  Chapter  I  for  two 
and  three  dimensions).  The  equations  of  motion  for  an  arbi- 
trary field  are 


Xn  = 


We  confine  ourselves  to  the  simplest  questions.  If  the  initial 
position  and  initial  direction  are  kept  fixed,  and  only  the  initial 
speed  v  is  varied,  what  are  the  properties  of  the  oo1  trajectories 
obtained?  The  simplest  geometric  result  is  that  rs  (the  rate  of 
variation  of  the  radius  of  curvature  with  respect  to  the  arc 
length)  varies  as  a  linear  function  of  r.  The  locus  of  the  centers 
of  the  osculating  spheres  is  a  straight  line,  just  as  in  the  case 
where  n  is  three. 


108  THE   PRINCETON   COLLOQUIUM. 

128.  A  general  curve  in  n-space  has  at  each  point  an  osculating 
plane,  an  osculating  3-flat,  and  so  on  up  to  an  osculating  (n—  1)- 
flat.     It  is  obvious  that  our  oo1  trajectories  have  the  same  os- 
culating plane  since  this  is  determined   by  the  given  initial 
direction  and  the  direction  of  the  force.     It  can  be  shown  that 
the  osculating  3-flat  is  also  fixed;  the  4-flat  varies,  generating  a 
pencil;  the  5-flat  varies,  generating  a  quadratic  system;  and  so 
on,  with  more  complicated  variations. 

129.  Consider  next  the  connection  between  the  various  cur- 
vatures and  the  speed. 

In  the  plane  (n  =  2)  there  is  only  one  curvature  71,  and  this 
varies  inversely  as  the  square  of  v. 

In  space  (n  =  3)  the  first  curvature  71  varies  as  above,  and 
the  second  curvature  or  torsion  yz  remains  fixed. 

If  n  =  4,  we  have  three  curvatures.     The  laws  for  71  and  72 
are  as  above,  while 

73  =  ci  -f  c2y~2, 

where  c\,  c2  are  constants  (depending  of  course  on  the  given 
initial  lineal  element). 

If  n  =  5,  we  have  71  =  av~~,  72  =  b  (these  forms  are  valid 
for  any  dimensions)  and 

d\  +  dzv*  +  e?304 


=  V  ci  +  c2tT2  + 


73  =  *v  ci  -h  czv  •  -f  W  -,          7,,  = 


+ 


If  n  =  6,  73  remains  the  same,  the  numerator  in  74  is  replaced 
by  the  square  root  of  a  polynomial  involving  v8,  and  75  is  given 
by  a  rational  formula. 

It  is  easy  to  write  down  the  general  formulas  for  the  n  —  1 
curvatures  in  n  space.  All  except  the  first,  second,  and  the 
last  are  irrational.  These  results  are  to  be  regarded  as  general- 
izations of  the  elementary  fact  (included  in  the  formula  for 
centrifugal  force  v*/r),  that  the  ordinary  curvature  varies  as  v~2. 

130.  By  eliminating  v  from  any  two  of  the  formulas,  we  can 
obtain  purely  geometric  results.  For  example,  in  space  of  four 
dimensions,  73  =  A  -f-  Byi,  where  A  and  B  depend  only  on  the 


ASPECTS   OF  DYNAMICS.  109 

common  initial  element.     But  in  higher  spaces 

73=  V^  +  flTl+C7i2. 

This  is  the  form  required  in  particular  in  the  application  to  the 
problem  of  three  bodies,  since  the  representative  space  has  nine 
dimensions. 

§§  131-132.     INTERACTING   PARTICLES  IN  THE  PLANE  AND  IN 

SPACE 

131.  We  consider  the  motion  of  n  +  1  particles,  denoted  by 
M,  MI,  •  •  • ,  Mn,  moving  in  the  plane  under  the  action  of  any 
forces  depending  on  the  position  of  the  particles.  The  dif- 
ferential equations  of  motion  are  then  of  the  form 

x  =  <p(x,  y,  xi,  yi,  •••,  xn,  yn], 

y  =  \l/(.r,  y,  Xi,  yi,  •••,  xn,  yn), 
xi  =  <Pi(x,  y,  *i,  ?/i,  •  •  •,  xn,  yn), 
i/i  =  ^i(ar,  y,  ari,  yi,  •  •-,  ar»,  yn), 

and  so  on,  where  the  masses — which  cannot  be  assumed  to  be 
unity  as  in  the  case  of  a  single  particle — are  absorbed  with  the 
forces  in  the  right  hand  terms.  From  these  equations  the  fol- 
lowing properties  may  be  deduced. 

(1)  Given  the  phases  of  MI,  •  •  • ,  and  the  position  and  the 
direction  of  M,  a  set  of  oo 1  trajectories  of  M  is  determined  (one 
for  each  value  of  the  speed).     The  foci  of  the  osculating  parabolas 
lie  on  a  special  quartic  curve  whose  inverse  with  respect  to  the 
given  point  is  a  parabola  tangent  to  the  given  initial  line  (the 
point  of  contact,  however,  is  usually  not  the  given  point). 

(2)  If  the  speed  of  one  of  the  remaining  particles,  say  MI,  is 
varied,   all   the   other  initial   conditions   being   unaltered,   the 
parabolic  locus  just  obtained  varies.     Its  point  of  contact  with 
the  initial  line  remains  fixed  and  all  the  oo1  parabolas,  one  for 
each  value  of  the  speed,  are  homothetic  with  respect  to  the  point 
of  tangency. 


110  THE   PRINCETON    COLLOQUIUM. 

(3)  The  normal  constructed  at  the  common  point  of  tangency 
cuts  the  parabola  again  at  a  distance  d  which  varies  in  such  a 
way  that  the  square  root  of  d  can  be  expressed  as  a  linear  com- 
bination of  the  square  roots  of  the  radii  of  curvature  of  the  cor- 
responding trajectories  described  by  the  particles  M\,  •  •  • ,  Mn. 

(4)  If  we  preserve  the  phases  of  the  particles  M\,  •  •  • ,  M nt 
then,  for  each  initial  direction  y'  of  M,  we  obtain,  by   (1),  a 
certain   parabolic   locus.     Consider   the   relation   between   the 
axis  of  this  parabola  and  the  initial  direction.     It  is  found  that 
the  initial  direction  y'  always  bisects  the  angle  between  the 
direction  of  the  force  acting  at  the  given  point  and  the  direction 
of  the  axis  of  the  parabola. 

(5)  Furthermore,  the  point  where  the  parabola  touches  the 
initial  line  describes,,  when  y'  varies,  a  quartic  curve  whose 
inverse  with  respect  to  the  given  point  is  a  conic  passing  through 
that  point  in  the  direction  of  the  force. 

It  is  to  be  observed  that  the  statement  (3)  about  the  variation 
of  d  simplifies  considerably  in  the  case  of  two  particles  (that  is, 
n  =  1).  In  that  case  d  varies  directly  as  the  radius  of  curva- 
ture of  the  trajectory  described  by  M\. 

132.  A  few  corresponding  results  for  the  case  of  any  number 
of  particles  moving  in  space  are  as  follows:  If  the  speed  of  M 
is  the  sole  arbitrary  parameter,  the  oo1  trajectories  of  M  have 
the  same  osculating  plane;  the  torsion  varies  according  to  a 
linear  integral  function  of  the  square  root  of  the  curvature;  the 
locus  of  the  centers  of  the  osculating  spheres  is  a  cubic  curve  of 
special  type. 

If  we  assign  the  phases  of  all  the  particles  except  M \  and  assign 
the  position  and  direction  of  3/i,  then  the  speed  of  MI,  or,  in 
consequence,  the  curvature  of  the  trajectory  described  by  M\, 
is  the  only  arbitrary  parameter.  There  will  then  be  « l  corre- 
sponding trajectories  described  by  M.  These  will  of  course  start 
from  the  same  point  in  the  same  direction  with  a  common  os- 
culating plane  and  a  common  curvature,  that  is,  they  all  have 
contact  of  the  second  order.  The  torsion  varies  and  so  does  the 


ASPECTS   OF   DYNAMICS.  Ill 

center  of  the  osculating  sphere.  The  simultaneous  variation  is 
controlled  by  the  law  that  the  distance  from  the  center  of  the  os- 
culating sphere  to  the  fixed  center  of  curvature  varies  as  a  linear 
integral  function  of  the  radius  of  torsion.  An  equivalent  state- 
ment is  that  the  rate  of  variation  of  the  radius  of  curvature 
per  unit  of  the  arc  is  expressed  by  a  linear  integral  function  of 
the  torsion. 

All  these  results  apply  in  particular  to  the  three-body  problem. 
The  present  application  is  more  concrete  than  that  indicated  in 
§  130,  since  no  higher  space  is  here  introduced.* 

§§  133-141.    FORCES  DEPENDING  ON  THE  TIME.    TRAJECTORIES 
AND  SPACE-TIME  CURVES 

133.  Hitherto  the  force  has  been  assumed  to  be  independent 
of  the  time;  now  we  consider  the  generalization  where  the  force 
depends  in  any  way  upon  the  time  as  well  as  the  position.  Take 
the  case  of  a  particle  moving  in  the  plane;  the  equations  of  motion 
are  then  of  the  form 

(1)  x  =  <p(x,  y,  0,        y  =  t(x,  y,  <)• 

From  these,  by  differentiation  and  elimination,  we  may  derive 

(2)  y'"  =  Py"  +  Qy"2  +  Ry"1, 

where  the  coefficients  are  functions  of  x,  y,  y',  t,  namely, 


p  = 


-  Y<P 

R  =  T 


If  we  are  given  the  initial  time,  position  and  direction,  that  is, 
the  initial  values  of  t,  x,  y,  y',  there  will  be  a  certain  set  of  <x> l 

*  Since  the  forces  in  the  three-body  problem  are  conservative,  we  may 
decompose  the  motions  into  natural  families,  and  interpret  each  family  in  a 
flat  space  of  eight  dimensions.  The  circles  of  curvature  at  a  given  point  will 
meet  again;  eight  of  them  will  be  hyperosculating,  and  these  will  be  mutually 
orthogonal.  Cf.  §  70. 


112  THE   PRINCETON   COLLOQUIUM. 

trajectories,  one  for  each  value  of  the  initial  speed.     The  follow- 
ing properties  are  obtained: 

(1)  We  find  that  the  focal  locus  (that  is,  the  locus  of  the  foci  of 
the  oo  l  osculating  parabolas)  is  a  quartic  curve  whose  inverse  with 
respect  to  the  given  point  is  a  parabola  which  is  tangent  to  the 
given  direction  line  (the  point  of  contact  is  not  usually  at  the 
given  point). 

(2)  As  y'  varies  (x,  y,  t  being  held  fixed)  this  point  of  contact 
describes  a  cubic  curve  whose  inverse  is  a  conic  passing  through 
the  given  point  in  the  direction  of  the  force. 

(3)  The  initial  direction  of  y'  bisects  the  angle  between  the 
direction  of  the  force  and  the  direction  of  the  axis  of  the  parabola 
described  in  (1). 

134.  The  total  system  of  trajectories,  for  all  initial  conditions, 
consists  of  oo4  curves.     Only  in  the  case  where  the  force  does 
not  depend  upon  the  time  does  the  system  consist  of  oo3  tra- 
jectories.    In  the  properties  stated  above,  the  initial  time  is 
kept  fixed.     In  a  certain  sense  then  the  results  are  not  purely 
geometric:  they  would  not  appear  in  a  photograph  of  the  complete 
system  of  trajectories.     This  system  will  be  represented  by  a  cer- 
tain differential  equation  of  the  fourth  order;  but  it  is  not  possible 
to  carry  out  the  requisite  eliminations  in  explicit  form,  and  hence 
the  derivation  of  purely  geometric  properties  involves  essentially 
new  difficulties.     A  complete  characterization  is  however  ob- 
tained, by  projection  from  space  curves,  in  §§  136,  140. 

135.  There  is  an  interesting  special  case  in  which  the  elimina- 
tion can  be  carried  out:  namely,  the  problem  of  the  motion  of  a 
particle  of  variable  mass  in  a  positional  field  of  force.     The  time 
then  appears  only  through  the  mass,  so  the  equations  of  motion 
are  of  the  form 

(3)  /(O*  =  v(x,  y}, 


As  the  result  of  the  elimination  is  complicated,  we  shall  here 
consider  only  the  case  where  the  function  f(t),  representing  the 
mass,  is  of  one  of  the  special  types  t4,  tz,  e',  (log  O2-  The  equa- 


ASPECTS    OF   DYNAMICS.  113 

tion  of  the  fourth  order  representing  the  trajectories  is  then 
found  to  be  of  the  form 

(4)  y™  =  Ay'"2  +  By"1  +  C, 

where  A,  B,  C  involve  only  x,  y,  y',  y". 

We  see  that  the  fourth  derivative  is  a  quadratic  function  of  the 
third  derivative.  This  category  of  equations  of  the  fourth  order 
arises  in  a  number  of  different  connections,  in  particular  in  the 
inverse  problem  of  the  calculus  of  variations,  as  stated  in  §  116. 
The  characteristic  geometric  property  may  in  the  present  case 
be  stated  as  follows: 

If  the  particle,  whose  mass  varies  according  to  one  of  the  four 
laws  stated,  is  projected  into  a  field  of  force  from  a  fixed  initial 
position  in  a  fixed  direction  at  different  times,  with  the  initial 
speed  for  each  time  so  adjusted  as  to  cause  the  initial  curvature 
of  the  trajectory  to  have  a  fixed  value,  and  if  for  each  of  the 
oo l  trajectories  thus  obtained  we  construct  the  osculating  conic 
(having  five-point  contact),  the  locus  of  the  centers  of  these 
conies  is  a  conic  passing  through  the  given  conic  in  the  given 
direction. 

Of  course  not  every  system  of  oo 4  curves  having  this  property 
can  be  regarded  as  a  trajectory  system  corresponding  to  equations 
of  motion  of  the  form  considered.  We  do  not,  however,  attempt 
a  complete  characterization. 

136.  Space-time  Curves. — When  we  integrate  the  equations 
of  motion,  either  in  the  special  case  where  the  forces  depend  only 
on  the  position 

(!')  x  =  <p(x,  y),        y  =  $(x,  y}, 

or  in  the  general  case  where  the  force  depends  also  on  the  time 

(1)  *  =  <p(x,  y,  t),      y  =  t(x,  y,  0, 

we  obtain  x  and  y  expressed  as  functions  of  t  and  four  constants 
of  integration.     If  we  represent  t  by  an  ordinate  perpendicular 
rfto  the  xy-p\sme,  thus  considering  x,  y,  t  as  rectangular  coordinates 
16 


114  THE  PRINCETON  COLLOQUIUM. 

in  space,  we  obtain  a  certain  system  of  oc  4  curves  in  that  space 
which  we  designate  as  space-time  curves.* 

If  we  project  these  curves  orthogonally  on  the  zz/-plane,  we 
obtain  the  trajectories.  In  the  general  case  (1)  there  will  be 
oo  4  of  these  trajectories;  but  in  the  special  case  where  the  force 
is  positional,  only  <x>3  trajectories  arise,  since  the  system  of  space- 
time  curves,  whose  number  is  still  oo  4,  now  admits  the  group  of 
translations  along  the  £-axis. 

If  we  project  the  space-time  curves  orthogonally  on  the  xt- 
plane  and  on  the  yt-p\ane,  we  obtain  in  each  case  a  system  of  oo  4 
plane  curves. 

What  are  the  properties  of  the  system  of  oo  4  space-time  curves? 
The  following  two  properties  are  characteristic: 

(1).  The  osculating  planes  of  the  oo  2  space-time  curves  through 
a  given  point  go  through  a  fixed  line  parallel  to  the  ary-plane. 
(This  line  is  parallel  to  the  direction  of  the  force  acting  at  the 
projected  point  in  the  xy-plane.) 

(2).  If  the  oo  2  space-time  curves  through  the  given  point  are 
orthogonally  projected  on  any  plane  perpendicular  to  the  in/- 
plane, the  oo  2  plane  curves  obtained  are  such  that  those  which 
have  the  same  tangent  also  have  the  same  curvature. 

Another  complete  characterization  may  be  given  as  follows: 

(3).  If  the  oo  2  space-time  curves  through  a  given  point  are 
orthogonally  projected  on  either  the  .rtf-plane  or  the  2/2-plane, 
the  oo  2  plane  curves  obtained  have  their  centers  of  curvature 
located  on  a  special  cubic  of  the  form  1?=a(x2-\-t2)  or  fl=b(y-+(i). 
A  corresponding  cubic  locus  will  then  necessarily  arise  by  pro- 
jection on  any  plane  perpendicular  to  the  a-y- 


*  It  may  be  remarked  that  if,  in  problem  (1),  the  force  is  multiplied  by  a 
constant  c  (or,  what  is  equivalent,  the  mass  of  the  particle  is  multiplied  by 
lie),  a  distinct  system  of  oo4  space-time  curves  will  be  obtained.  The  totality 
of  oo5  space  curves,  thus  related  to  the  oo1  plane  problems 

x  -  c<p(x,  y,  t),        y  =  ct(x,  y,  I), 


may  be  generated  as  trajectories  in  a  three-dimensional  positional  field  of 
force.  The  oo5  curves  have  the  four  characteristic  properties  of  a  space 
system  (§  11)  and  the  further  peculiarity  that  the  direction  of  the  force  is 
parallel  to  the  xy-plane. 


ASPECTS   OF  DYNAMICS.  115 

137.  Consider  the  oo4  curves  in  say  the  atf-plane.     These  are 
the  curves  representing  graphically  the  relation  between  the 
abscissa  x  and  the  time  t.     By  eliminating  y  from  the  set  (1), 
we  obtain  a  relation  of  the  form 

x™  =  Ax*  +  Bx+C, 

where  A,  B,  C  involve  only  x,  x,  x  and  the  independent  variable 
t.  The  fourth  derivative  is  thus  always  quadratic  with  respect 
to  the  third  derivative.  Hence,  by  §  116,  we  have  this  result: 

In  the  xt-plane  (or,  more  generally,  in  any  plane  perpendicular 
to  the  plane  xy  in  which  the  motion  actually  takes  place),  the 
oo l  curves  having  any  element  of  curvature  in  common  are  such 
that  the  locus  of  the  centers  C'  of  their  osculating  conies  (con- 
structed at  the  common  point)  is  a  conic  passing  through  the 
common  point  in  the  direction  of  the  common  tangent. 

As  indicated  above,  the  oo 4  curves  in  the  xy-plane,  that  is,  the 
trajectories,  do  not  usually  enjoy  this  simple  property.  Even 
in  the  case  where  the  time  enters  only  through  the  mass,  the 
locus  of  the  centers  of  the  osculating  conies  may  be  of  any 
degree  of  complication.  Its  shape  depends  on  the  law  of  vari- 
ation of  the  mass.  Only  for  the  special  laws  stated  at  the  bottom 
of  page  112,  together  with  certain  combinations  of  them,  is  the 
equation  of  the  trajectories  of  the  quadratic  type. 

138.  It  is  possible  to  obtain  additional  general  properties  of  the 
.rt-system,  describing  how  the  locus  conic,  corresponding  to  a  cur- 
vature element,  changes  wrhen  the  element  changes.     For  the  co- 
efficients A,  B,  C  determining  the  position  of  the  conic  have  the 
following  forms  :  A  does  not  involve  x,  B  is  linear  and  integral  in 
x,  C  is  quadratic  and  integral  in  x.     Hence  these  results: 

If  the  curvature  element  is  varied,  at  the  given  point  0,  in  such 
a  way  that  the  second  derivative  x  is  constant,  so  that  only  x 
varies,  the  center  C"  of  the  corresponding  locus  conic  describes 
a  new  conic. 

At  the  same  time  a  certain  two-to-one  correspondence  arises 
between  the  initial  direction  of  the  element  and  the  direction  of 
the  line  joining  0  to  the  center  C". 


116  THE   PRINCETON  COLLOQUIUM. 

139.  A  clearer  picture  is  perhaps  obtained  by  changing  the 
'notation  to  correspond  with  the  usual  x,  y,  z  notation  for  rec- 
tangular coordinates  in  space.  It  is  then  desirable  to  lay  off 
the  time  on  the  rr-axis,  since  this  is  the  independent  variable. 
The  actual  motion  then  takes  place  in  the  7/z-plane,  and  the 
differential  equations  of  motion  are 

d?y  d?z 

jy?  =  *>(*»  y>  2)>      ^  =  vfa  y>  2)- 

The  curves  in  space  x,  y,  z  are  then  the  space-time  curves.  Their 
projections  on  the  i/z-plane  are  the  trajectories  (whose  explicit 
properties  have  not  been  derived).  Their  projections  on  the  xy- 
plane  (or  on  the  o-z-plane,  or  on  any  plane  parallel  to  the  z-axis) 
are  curves  whose  properties  have  just  been  stated  (§§  137,  138). 
The  differential  equation  in  the  xy-p\ane  is 


where  the  coefficients  involve  only  x,  y,  and  y1  '. 

140.  We  have  not  attempted  a  complete  direct  characterization 
of  the  systems  of  curves  arising  in  any  one  of  the  coordinate 
planes.     Such  a  characterization  has  however  been  given  (§  136) 
for  the  system  of  <x>  4  space-time  curves.     Indirectly  this  really 
solves  all  the  problems.     A  system  of  curves  in  the  plane  can  be 
regarded  as  trajectories  of  a  force  depending  on  time  and  position 
if  and  only  if  the  curves  can  be  obtained  by  orthogonal  projection 
from  some  system  of  oo4  curves  in  space  having  the  properties  (1) 
and  (2)  of  §  136.     If,  furthermore,  the  space  system  is  invariant 
under  translation  perpendicular  to  the  given  plane,  the  plane 
system,  then  consisting  of  only  oo3  curves,  belongs  to  a  posi- 
tional field. 

141.  For  any  force  depending  on  time  and  position 

x  =  <p(x,  y,  t),        y  =  \f/(x,  y,  t), 
the  number  of  space-time  curves  is  always  oo  4.     When  we  project 


ASPECTS   OF  DYNAMICS.  117 

these  on  the  .ry-plane,  to  obtain  the  trajectories,  the  number  is 
usually  oo 4.  The  number  reduces  to  oo 3  if  the  force  is  positional 
but  does  not  vanish ;  in  the  latter  case  the  trajectories  are  merely 
the  oo 2  straight  lines. 

In  the  xt-p\ane  the  usual  number  of  curves  is  oo4.  The  only 
exception  arises  when  the  function  (p  is  free  from  the  variable 
y.  In  this  case  the  ^-curves  all  satisfy  the  equation  of  second 
order  x  =  <p(x,  t)  and  therefore  their  number  is  only  oo2.  Similar 
statements  hold  of  course  for  the  yt-plane. 

Consider,  as  a  single  example,  gravity,  taken  as  uniform  and 
acting  in  the  vertical  ary-plane.  The  equations  of  motion  are 

x  =  0,        y  =  g. 
The  xyt-curves  are 

x  =  at  +  b,        y  =  \g$  +  ct  +  d, 

a  certain  family  of  oo4  parabolas  in  space.  The  atf-curves  are 
oo 2  straight  lines.  The  yt-curves  are  oo2  parabolas.  The  xy- 
curves  (that  is,  the  trajectories)  are  °o3  parabolas 

y  =  ax"2  +  /to2  +  7. 

It  is  to  be  observed  that  if  the  gravity  constant  g  is  changed, 
the  new  problem,  while  giving  the  same  trajectories,  gives  a  dis- 
tinct family  of  xyt-curves.  If  g  takes  all  possible  values,  the 
totality  of  space-time  curves  obtained  is  formed  of  oo 5  parabolas 
(namely,  those  whose  axes  are  parallel  to  the  2-axis).  These 
curves,  in  accordance  with  the  general  statement  made  in  the 
footnote  on  page  114,  are  the  trajectories  of  a  positional  field 
in  space,  the  generating  force  being  constant  and  acting  in  the 
^-direction. 

All  the  results  can  be  extended  so  as  to  apply  to  the  four- 
dimensional  space-time  curves  depicting  motion  in  ordinary 
space. 


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